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Theorem setcepi 17979
Description: An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcmon.c 𝐢 = (SetCatβ€˜π‘ˆ)
setcmon.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
setcmon.x (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
setcmon.y (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
setcepi.h 𝐸 = (Epiβ€˜πΆ)
setcepi.2 (πœ‘ β†’ 2o ∈ π‘ˆ)
Assertion
Ref Expression
setcepi (πœ‘ β†’ (𝐹 ∈ (π‘‹πΈπ‘Œ) ↔ 𝐹:𝑋–ontoβ†’π‘Œ))

Proof of Theorem setcepi
Dummy variables π‘₯ 𝑔 π‘Ž β„Ž 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
2 eqid 2733 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
3 eqid 2733 . . . . . 6 (compβ€˜πΆ) = (compβ€˜πΆ)
4 setcepi.h . . . . . 6 𝐸 = (Epiβ€˜πΆ)
5 setcmon.u . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
6 setcmon.c . . . . . . . 8 𝐢 = (SetCatβ€˜π‘ˆ)
76setccat 17976 . . . . . . 7 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
85, 7syl 17 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ Cat)
9 setcmon.x . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
106, 5setcbas 17969 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΆ))
119, 10eleqtrd 2836 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜πΆ))
12 setcmon.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
1312, 10eleqtrd 2836 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (Baseβ€˜πΆ))
141, 2, 3, 4, 8, 11, 13epihom 17630 . . . . 5 (πœ‘ β†’ (π‘‹πΈπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
1514sselda 3945 . . . 4 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
166, 5, 2, 9, 12elsetchom 17972 . . . . 5 (πœ‘ β†’ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
1716biimpa 478 . . . 4 ((πœ‘ ∧ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
1815, 17syldan 592 . . 3 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
1918frnd 6677 . . . 4 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ran 𝐹 βŠ† π‘Œ)
2018ffnd 6670 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐹 Fn 𝑋)
21 fnfvelrn 7032 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹)
2220, 21sylan 581 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) ∧ π‘₯ ∈ 𝑋) β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹)
2322iftrued 4495 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) ∧ π‘₯ ∈ 𝑋) β†’ if((πΉβ€˜π‘₯) ∈ ran 𝐹, 1o, βˆ…) = 1o)
2423mpteq2dva 5206 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘₯ ∈ 𝑋 ↦ if((πΉβ€˜π‘₯) ∈ ran 𝐹, 1o, βˆ…)) = (π‘₯ ∈ 𝑋 ↦ 1o))
2518ffvelcdmda 7036 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) ∧ π‘₯ ∈ 𝑋) β†’ (πΉβ€˜π‘₯) ∈ π‘Œ)
2618feqmptd 6911 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜π‘₯)))
27 eqidd 2734 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)))
28 eleq1 2822 . . . . . . . . . . . . 13 (π‘Ž = (πΉβ€˜π‘₯) β†’ (π‘Ž ∈ ran 𝐹 ↔ (πΉβ€˜π‘₯) ∈ ran 𝐹))
2928ifbid 4510 . . . . . . . . . . . 12 (π‘Ž = (πΉβ€˜π‘₯) β†’ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = if((πΉβ€˜π‘₯) ∈ ran 𝐹, 1o, βˆ…))
3025, 26, 27, 29fmptco 7076 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) ∘ 𝐹) = (π‘₯ ∈ 𝑋 ↦ if((πΉβ€˜π‘₯) ∈ ran 𝐹, 1o, βˆ…)))
31 fconstmpt 5695 . . . . . . . . . . . . 13 (π‘Œ Γ— {1o}) = (π‘Ž ∈ π‘Œ ↦ 1o)
3231a1i 11 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Œ Γ— {1o}) = (π‘Ž ∈ π‘Œ ↦ 1o))
33 eqidd 2734 . . . . . . . . . . . 12 (π‘Ž = (πΉβ€˜π‘₯) β†’ 1o = 1o)
3425, 26, 32, 33fmptco 7076 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Œ Γ— {1o}) ∘ 𝐹) = (π‘₯ ∈ 𝑋 ↦ 1o))
3524, 30, 343eqtr4d 2783 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) ∘ 𝐹) = ((π‘Œ Γ— {1o}) ∘ 𝐹))
365adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ π‘ˆ ∈ 𝑉)
379adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝑋 ∈ π‘ˆ)
3812adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ π‘Œ ∈ π‘ˆ)
39 setcepi.2 . . . . . . . . . . . 12 (πœ‘ β†’ 2o ∈ π‘ˆ)
4039adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 2o ∈ π‘ˆ)
41 eqid 2733 . . . . . . . . . . . . 13 (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…))
42 1oex 8423 . . . . . . . . . . . . . . . . 17 1o ∈ V
4342prid2 4725 . . . . . . . . . . . . . . . 16 1o ∈ {βˆ…, 1o}
44 df2o3 8421 . . . . . . . . . . . . . . . 16 2o = {βˆ…, 1o}
4543, 44eleqtrri 2833 . . . . . . . . . . . . . . 15 1o ∈ 2o
46 0ex 5265 . . . . . . . . . . . . . . . . 17 βˆ… ∈ V
4746prid1 4724 . . . . . . . . . . . . . . . 16 βˆ… ∈ {βˆ…, 1o}
4847, 44eleqtrri 2833 . . . . . . . . . . . . . . 15 βˆ… ∈ 2o
4945, 48ifcli 4534 . . . . . . . . . . . . . 14 if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) ∈ 2o
5049a1i 11 . . . . . . . . . . . . 13 (π‘Ž ∈ π‘Œ β†’ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) ∈ 2o)
5141, 50fmpti 7061 . . . . . . . . . . . 12 (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)):π‘ŒβŸΆ2o
5251a1i 11 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)):π‘ŒβŸΆ2o)
536, 36, 3, 37, 38, 40, 18, 52setcco 17974 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…))(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)2o)𝐹) = ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) ∘ 𝐹))
54 fconst6g 6732 . . . . . . . . . . . 12 (1o ∈ 2o β†’ (π‘Œ Γ— {1o}):π‘ŒβŸΆ2o)
5545, 54mp1i 13 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Œ Γ— {1o}):π‘ŒβŸΆ2o)
566, 36, 3, 37, 38, 40, 18, 55setcco 17974 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Œ Γ— {1o})(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)2o)𝐹) = ((π‘Œ Γ— {1o}) ∘ 𝐹))
5735, 53, 563eqtr4d 2783 . . . . . . . . 9 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…))(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)2o)𝐹) = ((π‘Œ Γ— {1o})(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)2o)𝐹))
588adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐢 ∈ Cat)
5911adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝑋 ∈ (Baseβ€˜πΆ))
6013adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ π‘Œ ∈ (Baseβ€˜πΆ))
6139, 10eleqtrd 2836 . . . . . . . . . . 11 (πœ‘ β†’ 2o ∈ (Baseβ€˜πΆ))
6261adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 2o ∈ (Baseβ€˜πΆ))
63 simpr 486 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐹 ∈ (π‘‹πΈπ‘Œ))
646, 36, 2, 38, 40elsetchom 17972 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) ∈ (π‘Œ(Hom β€˜πΆ)2o) ↔ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)):π‘ŒβŸΆ2o))
6552, 64mpbird 257 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) ∈ (π‘Œ(Hom β€˜πΆ)2o))
666, 36, 2, 38, 40elsetchom 17972 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ((π‘Œ Γ— {1o}) ∈ (π‘Œ(Hom β€˜πΆ)2o) ↔ (π‘Œ Γ— {1o}):π‘ŒβŸΆ2o))
6755, 66mpbird 257 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Œ Γ— {1o}) ∈ (π‘Œ(Hom β€˜πΆ)2o))
681, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67epii 17631 . . . . . . . . 9 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…))(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)2o)𝐹) = ((π‘Œ Γ— {1o})(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)2o)𝐹) ↔ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Œ Γ— {1o})))
6957, 68mpbid 231 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Œ Γ— {1o}))
7069, 31eqtrdi 2789 . . . . . . 7 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ (π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Ž ∈ π‘Œ ↦ 1o))
7149rgenw 3065 . . . . . . . 8 βˆ€π‘Ž ∈ π‘Œ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) ∈ 2o
72 mpteqb 6968 . . . . . . . 8 (βˆ€π‘Ž ∈ π‘Œ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) ∈ 2o β†’ ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Ž ∈ π‘Œ ↦ 1o) ↔ βˆ€π‘Ž ∈ π‘Œ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o))
7371, 72ax-mp 5 . . . . . . 7 ((π‘Ž ∈ π‘Œ ↦ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…)) = (π‘Ž ∈ π‘Œ ↦ 1o) ↔ βˆ€π‘Ž ∈ π‘Œ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o)
7470, 73sylib 217 . . . . . 6 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ βˆ€π‘Ž ∈ π‘Œ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o)
75 1n0 8435 . . . . . . . . . 10 1o β‰  βˆ…
7675nesymi 2998 . . . . . . . . 9 Β¬ βˆ… = 1o
77 iffalse 4496 . . . . . . . . . 10 (Β¬ π‘Ž ∈ ran 𝐹 β†’ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = βˆ…)
7877eqeq1d 2735 . . . . . . . . 9 (Β¬ π‘Ž ∈ ran 𝐹 β†’ (if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o ↔ βˆ… = 1o))
7976, 78mtbiri 327 . . . . . . . 8 (Β¬ π‘Ž ∈ ran 𝐹 β†’ Β¬ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o)
8079con4i 114 . . . . . . 7 (if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o β†’ π‘Ž ∈ ran 𝐹)
8180ralimi 3083 . . . . . 6 (βˆ€π‘Ž ∈ π‘Œ if(π‘Ž ∈ ran 𝐹, 1o, βˆ…) = 1o β†’ βˆ€π‘Ž ∈ π‘Œ π‘Ž ∈ ran 𝐹)
8274, 81syl 17 . . . . 5 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ βˆ€π‘Ž ∈ π‘Œ π‘Ž ∈ ran 𝐹)
83 dfss3 3933 . . . . 5 (π‘Œ βŠ† ran 𝐹 ↔ βˆ€π‘Ž ∈ π‘Œ π‘Ž ∈ ran 𝐹)
8482, 83sylibr 233 . . . 4 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ π‘Œ βŠ† ran 𝐹)
8519, 84eqssd 3962 . . 3 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ ran 𝐹 = π‘Œ)
86 dffo2 6761 . . 3 (𝐹:𝑋–ontoβ†’π‘Œ ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ ran 𝐹 = π‘Œ))
8718, 85, 86sylanbrc 584 . 2 ((πœ‘ ∧ 𝐹 ∈ (π‘‹πΈπ‘Œ)) β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
88 fof 6757 . . . . 5 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
8988adantl 483 . . . 4 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
9016biimpar 479 . . . 4 ((πœ‘ ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
9189, 90syldan 592 . . 3 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
9210adantr 482 . . . . . 6 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ π‘ˆ = (Baseβ€˜πΆ))
9392eleq2d 2820 . . . . 5 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝑧 ∈ π‘ˆ ↔ 𝑧 ∈ (Baseβ€˜πΆ)))
945ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ π‘ˆ ∈ 𝑉)
959ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝑋 ∈ π‘ˆ)
9612ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ π‘Œ ∈ π‘ˆ)
97 simprl 770 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝑧 ∈ π‘ˆ)
9889adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
99 simprrl 780 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧))
1006, 94, 2, 96, 97elsetchom 17972 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ↔ 𝑔:π‘ŒβŸΆπ‘§))
10199, 100mpbid 231 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝑔:π‘ŒβŸΆπ‘§)
1026, 94, 3, 95, 96, 97, 98, 101setcco 17974 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ (𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (𝑔 ∘ 𝐹))
103 simprrr 781 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧))
1046, 94, 2, 96, 97elsetchom 17972 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ (β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ↔ β„Ž:π‘ŒβŸΆπ‘§))
105103, 104mpbid 231 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ β„Ž:π‘ŒβŸΆπ‘§)
1066, 94, 3, 95, 96, 97, 98, 105setcco 17974 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž ∘ 𝐹))
107102, 106eqeq12d 2749 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ ((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) ↔ (𝑔 ∘ 𝐹) = (β„Ž ∘ 𝐹)))
108 simplr 768 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
109101ffnd 6670 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ 𝑔 Fn π‘Œ)
110105ffnd 6670 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ β„Ž Fn π‘Œ)
111 cocan2 7239 . . . . . . . . . . 11 ((𝐹:𝑋–ontoβ†’π‘Œ ∧ 𝑔 Fn π‘Œ ∧ β„Ž Fn π‘Œ) β†’ ((𝑔 ∘ 𝐹) = (β„Ž ∘ 𝐹) ↔ 𝑔 = β„Ž))
112108, 109, 110, 111syl3anc 1372 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ ((𝑔 ∘ 𝐹) = (β„Ž ∘ 𝐹) ↔ 𝑔 = β„Ž))
113112biimpd 228 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ ((𝑔 ∘ 𝐹) = (β„Ž ∘ 𝐹) β†’ 𝑔 = β„Ž))
114107, 113sylbid 239 . . . . . . . 8 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)))) β†’ ((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž))
115114anassrs 469 . . . . . . 7 ((((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝑧 ∈ π‘ˆ) ∧ (𝑔 ∈ (π‘Œ(Hom β€˜πΆ)𝑧) ∧ β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧))) β†’ ((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž))
116115ralrimivva 3194 . . . . . 6 (((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) ∧ 𝑧 ∈ π‘ˆ) β†’ βˆ€π‘” ∈ (π‘Œ(Hom β€˜πΆ)𝑧)βˆ€β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž))
117116ex 414 . . . . 5 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝑧 ∈ π‘ˆ β†’ βˆ€π‘” ∈ (π‘Œ(Hom β€˜πΆ)𝑧)βˆ€β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž)))
11893, 117sylbird 260 . . . 4 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝑧 ∈ (Baseβ€˜πΆ) β†’ βˆ€π‘” ∈ (π‘Œ(Hom β€˜πΆ)𝑧)βˆ€β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž)))
119118ralrimiv 3139 . . 3 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ βˆ€π‘§ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (π‘Œ(Hom β€˜πΆ)𝑧)βˆ€β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž))
1201, 2, 3, 4, 8, 11, 13isepi2 17629 . . . 4 (πœ‘ β†’ (𝐹 ∈ (π‘‹πΈπ‘Œ) ↔ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ βˆ€π‘§ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (π‘Œ(Hom β€˜πΆ)𝑧)βˆ€β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž))))
121120adantr 482 . . 3 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐹 ∈ (π‘‹πΈπ‘Œ) ↔ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ βˆ€π‘§ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (π‘Œ(Hom β€˜πΆ)𝑧)βˆ€β„Ž ∈ (π‘Œ(Hom β€˜πΆ)𝑧)((𝑔(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) = (β„Ž(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑧)𝐹) β†’ 𝑔 = β„Ž))))
12291, 119, 121mpbir2and 712 . 2 ((πœ‘ ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ 𝐹 ∈ (π‘‹πΈπ‘Œ))
12387, 122impbida 800 1 (πœ‘ β†’ (𝐹 ∈ (π‘‹πΈπ‘Œ) ↔ 𝐹:𝑋–ontoβ†’π‘Œ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487  {csn 4587  {cpr 4589  βŸ¨cop 4593   ↦ cmpt 5189   Γ— cxp 5632  ran crn 5635   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358  1oc1o 8406  2oc2o 8407  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549  Epicepi 17617  SetCatcsetc 17966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-oppc 17597  df-mon 17618  df-epi 17619  df-setc 17967
This theorem is referenced by: (None)
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