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Theorem setcmon 18046
Description: A monomorphism of sets is an injection. (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)
setcmon.h 𝑀 = (Monoβ€˜πΆ)
Assertion
Ref Expression
setcmon (πœ‘ β†’ (𝐹 ∈ (π‘‹π‘€π‘Œ) ↔ 𝐹:𝑋–1-1β†’π‘Œ))

Proof of Theorem setcmon
Dummy variables π‘₯ 𝑔 β„Ž 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
2 eqid 2726 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
3 eqid 2726 . . . . . 6 (compβ€˜πΆ) = (compβ€˜πΆ)
4 setcmon.h . . . . . 6 𝑀 = (Monoβ€˜πΆ)
5 setcmon.u . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
6 setcmon.c . . . . . . . 8 𝐢 = (SetCatβ€˜π‘ˆ)
76setccat 18044 . . . . . . 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 18037 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΆ))
119, 10eleqtrd 2829 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜πΆ))
12 setcmon.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
1312, 10eleqtrd 2829 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (Baseβ€˜πΆ))
141, 2, 3, 4, 8, 11, 13monhom 17688 . . . . 5 (πœ‘ β†’ (π‘‹π‘€π‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
1514sselda 3977 . . . 4 ((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
166, 5, 2, 9, 12elsetchom 18040 . . . . 5 (πœ‘ β†’ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
1716biimpa 476 . . . 4 ((πœ‘ ∧ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
1815, 17syldan 590 . . 3 ((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
19 simprr 770 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
2019sneqd 4635 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ {(πΉβ€˜π‘₯)} = {(πΉβ€˜π‘¦)})
2120xpeq2d 5699 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝑋 Γ— {(πΉβ€˜π‘₯)}) = (𝑋 Γ— {(πΉβ€˜π‘¦)}))
2218adantr 480 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
2322ffnd 6711 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝐹 Fn 𝑋)
24 simprll 776 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ π‘₯ ∈ 𝑋)
25 fcoconst 7127 . . . . . . . . . . 11 ((𝐹 Fn 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (𝐹 ∘ (𝑋 Γ— {π‘₯})) = (𝑋 Γ— {(πΉβ€˜π‘₯)}))
2623, 24, 25syl2anc 583 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝐹 ∘ (𝑋 Γ— {π‘₯})) = (𝑋 Γ— {(πΉβ€˜π‘₯)}))
27 simprlr 777 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝑦 ∈ 𝑋)
28 fcoconst 7127 . . . . . . . . . . 11 ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝐹 ∘ (𝑋 Γ— {𝑦})) = (𝑋 Γ— {(πΉβ€˜π‘¦)}))
2923, 27, 28syl2anc 583 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝐹 ∘ (𝑋 Γ— {𝑦})) = (𝑋 Γ— {(πΉβ€˜π‘¦)}))
3021, 26, 293eqtr4d 2776 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝐹 ∘ (𝑋 Γ— {π‘₯})) = (𝐹 ∘ (𝑋 Γ— {𝑦})))
315ad2antrr 723 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ π‘ˆ ∈ 𝑉)
329ad2antrr 723 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝑋 ∈ π‘ˆ)
3312ad2antrr 723 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ π‘Œ ∈ π‘ˆ)
34 fconst6g 6773 . . . . . . . . . . 11 (π‘₯ ∈ 𝑋 β†’ (𝑋 Γ— {π‘₯}):π‘‹βŸΆπ‘‹)
3524, 34syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝑋 Γ— {π‘₯}):π‘‹βŸΆπ‘‹)
366, 31, 3, 32, 32, 33, 35, 22setcco 18042 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝐹(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)(𝑋 Γ— {π‘₯})) = (𝐹 ∘ (𝑋 Γ— {π‘₯})))
37 fconst6g 6773 . . . . . . . . . . 11 (𝑦 ∈ 𝑋 β†’ (𝑋 Γ— {𝑦}):π‘‹βŸΆπ‘‹)
3827, 37syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝑋 Γ— {𝑦}):π‘‹βŸΆπ‘‹)
396, 31, 3, 32, 32, 33, 38, 22setcco 18042 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝐹(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)(𝑋 Γ— {𝑦})) = (𝐹 ∘ (𝑋 Γ— {𝑦})))
4030, 36, 393eqtr4d 2776 . . . . . . . 8 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝐹(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)(𝑋 Γ— {π‘₯})) = (𝐹(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)(𝑋 Γ— {𝑦})))
418ad2antrr 723 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝐢 ∈ Cat)
4211ad2antrr 723 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝑋 ∈ (Baseβ€˜πΆ))
4313ad2antrr 723 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ π‘Œ ∈ (Baseβ€˜πΆ))
44 simplr 766 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ 𝐹 ∈ (π‘‹π‘€π‘Œ))
456, 31, 2, 32, 32elsetchom 18040 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ ((𝑋 Γ— {π‘₯}) ∈ (𝑋(Hom β€˜πΆ)𝑋) ↔ (𝑋 Γ— {π‘₯}):π‘‹βŸΆπ‘‹))
4635, 45mpbird 257 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝑋 Γ— {π‘₯}) ∈ (𝑋(Hom β€˜πΆ)𝑋))
476, 31, 2, 32, 32elsetchom 18040 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ ((𝑋 Γ— {𝑦}) ∈ (𝑋(Hom β€˜πΆ)𝑋) ↔ (𝑋 Γ— {𝑦}):π‘‹βŸΆπ‘‹))
4838, 47mpbird 257 . . . . . . . . 9 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝑋 Γ— {𝑦}) ∈ (𝑋(Hom β€˜πΆ)𝑋))
491, 2, 3, 4, 41, 42, 43, 42, 44, 46, 48moni 17689 . . . . . . . 8 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ ((𝐹(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)(𝑋 Γ— {π‘₯})) = (𝐹(βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)(𝑋 Γ— {𝑦})) ↔ (𝑋 Γ— {π‘₯}) = (𝑋 Γ— {𝑦})))
5040, 49mpbid 231 . . . . . . 7 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (𝑋 Γ— {π‘₯}) = (𝑋 Γ— {𝑦}))
5150fveq1d 6886 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ ((𝑋 Γ— {π‘₯})β€˜π‘₯) = ((𝑋 Γ— {𝑦})β€˜π‘₯))
52 vex 3472 . . . . . . . 8 π‘₯ ∈ V
5352fvconst2 7200 . . . . . . 7 (π‘₯ ∈ 𝑋 β†’ ((𝑋 Γ— {π‘₯})β€˜π‘₯) = π‘₯)
5424, 53syl 17 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ ((𝑋 Γ— {π‘₯})β€˜π‘₯) = π‘₯)
55 vex 3472 . . . . . . . 8 𝑦 ∈ V
5655fvconst2 7200 . . . . . . 7 (π‘₯ ∈ 𝑋 β†’ ((𝑋 Γ— {𝑦})β€˜π‘₯) = 𝑦)
5724, 56syl 17 . . . . . 6 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ ((𝑋 Γ— {𝑦})β€˜π‘₯) = 𝑦)
5851, 54, 573eqtr3d 2774 . . . . 5 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ π‘₯ = 𝑦)
5958expr 456 . . . 4 (((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦))
6059ralrimivva 3194 . . 3 ((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦))
61 dff13 7249 . . 3 (𝐹:𝑋–1-1β†’π‘Œ ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦)))
6218, 60, 61sylanbrc 582 . 2 ((πœ‘ ∧ 𝐹 ∈ (π‘‹π‘€π‘Œ)) β†’ 𝐹:𝑋–1-1β†’π‘Œ)
63 f1f 6780 . . . 4 (𝐹:𝑋–1-1β†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
6416biimpar 477 . . . 4 ((πœ‘ ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
6563, 64sylan2 592 . . 3 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
6610adantr 480 . . . . . 6 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ π‘ˆ = (Baseβ€˜πΆ))
6766eleq2d 2813 . . . . 5 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ (𝑧 ∈ π‘ˆ ↔ 𝑧 ∈ (Baseβ€˜πΆ)))
685ad2antrr 723 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ π‘ˆ ∈ 𝑉)
69 simprl 768 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ 𝑧 ∈ π‘ˆ)
709ad2antrr 723 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ 𝑋 ∈ π‘ˆ)
7112ad2antrr 723 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ π‘Œ ∈ π‘ˆ)
72 simprrl 778 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ 𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋))
736, 68, 2, 69, 70elsetchom 18040 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ↔ 𝑔:π‘§βŸΆπ‘‹))
7472, 73mpbid 231 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ 𝑔:π‘§βŸΆπ‘‹)
7563ad2antlr 724 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
766, 68, 3, 69, 70, 71, 74, 75setcco 18042 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹 ∘ 𝑔))
77 simprrr 779 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋))
786, 68, 2, 69, 70elsetchom 18040 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ (β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋) ↔ β„Ž:π‘§βŸΆπ‘‹))
7977, 78mpbid 231 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ β„Ž:π‘§βŸΆπ‘‹)
806, 68, 3, 69, 70, 71, 79, 75setcco 18042 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) = (𝐹 ∘ β„Ž))
8176, 80eqeq12d 2742 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ ((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) ↔ (𝐹 ∘ 𝑔) = (𝐹 ∘ β„Ž)))
82 simplr 766 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ 𝐹:𝑋–1-1β†’π‘Œ)
83 cocan1 7284 . . . . . . . . . . 11 ((𝐹:𝑋–1-1β†’π‘Œ ∧ 𝑔:π‘§βŸΆπ‘‹ ∧ β„Ž:π‘§βŸΆπ‘‹) β†’ ((𝐹 ∘ 𝑔) = (𝐹 ∘ β„Ž) ↔ 𝑔 = β„Ž))
8482, 74, 79, 83syl3anc 1368 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ ((𝐹 ∘ 𝑔) = (𝐹 ∘ β„Ž) ↔ 𝑔 = β„Ž))
8584biimpd 228 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ ((𝐹 ∘ 𝑔) = (𝐹 ∘ β„Ž) β†’ 𝑔 = β„Ž))
8681, 85sylbid 239 . . . . . . . 8 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ (𝑧 ∈ π‘ˆ ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)))) β†’ ((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
8786anassrs 467 . . . . . . 7 ((((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ 𝑧 ∈ π‘ˆ) ∧ (𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋))) β†’ ((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
8887ralrimivva 3194 . . . . . 6 (((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) ∧ 𝑧 ∈ π‘ˆ) β†’ βˆ€π‘” ∈ (𝑧(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
8988ex 412 . . . . 5 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ (𝑧 ∈ π‘ˆ β†’ βˆ€π‘” ∈ (𝑧(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž)))
9067, 89sylbird 260 . . . 4 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ (𝑧 ∈ (Baseβ€˜πΆ) β†’ βˆ€π‘” ∈ (𝑧(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž)))
9190ralrimiv 3139 . . 3 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ βˆ€π‘§ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (𝑧(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
921, 2, 3, 4, 8, 11, 13ismon2 17687 . . . 4 (πœ‘ β†’ (𝐹 ∈ (π‘‹π‘€π‘Œ) ↔ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ βˆ€π‘§ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (𝑧(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))))
9392adantr 480 . . 3 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ (𝐹 ∈ (π‘‹π‘€π‘Œ) ↔ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ βˆ€π‘§ ∈ (Baseβ€˜πΆ)βˆ€π‘” ∈ (𝑧(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (𝑧(Hom β€˜πΆ)𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))))
9465, 91, 93mpbir2and 710 . 2 ((πœ‘ ∧ 𝐹:𝑋–1-1β†’π‘Œ) β†’ 𝐹 ∈ (π‘‹π‘€π‘Œ))
9562, 94impbida 798 1 (πœ‘ β†’ (𝐹 ∈ (π‘‹π‘€π‘Œ) ↔ 𝐹:𝑋–1-1β†’π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {csn 4623  βŸ¨cop 4629   Γ— cxp 5667   ∘ ccom 5673   Fn wfn 6531  βŸΆwf 6532  β€“1-1β†’wf1 6533  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  Hom chom 17214  compcco 17215  Catccat 17614  Monocmon 17681  SetCatcsetc 18034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17151  df-hom 17227  df-cco 17228  df-cat 17618  df-cid 17619  df-mon 17683  df-setc 18035
This theorem is referenced by: (None)
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