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Theorem setcepi 16503
Description: An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcmon.c 𝐶 = (SetCat‘𝑈)
setcmon.u (𝜑𝑈𝑉)
setcmon.x (𝜑𝑋𝑈)
setcmon.y (𝜑𝑌𝑈)
setcepi.h 𝐸 = (Epi‘𝐶)
setcepi.2 (𝜑 → 2𝑜𝑈)
Assertion
Ref Expression
setcepi (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))

Proof of Theorem setcepi
Dummy variables 𝑥 𝑔 𝑎 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2605 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2605 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2605 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 setcepi.h . . . . . 6 𝐸 = (Epi‘𝐶)
5 setcmon.u . . . . . . 7 (𝜑𝑈𝑉)
6 setcmon.c . . . . . . . 8 𝐶 = (SetCat‘𝑈)
76setccat 16500 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 16493 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2685 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2685 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13epihom 16167 . . . . 5 (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3563 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 16496 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 499 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 485 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋𝑌)
19 frn 5948 . . . . 5 (𝐹:𝑋𝑌 → ran 𝐹𝑌)
2018, 19syl 17 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹𝑌)
21 ffn 5940 . . . . . . . . . . . . . . 15 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
2218, 21syl 17 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
23 fnfvelrn 6245 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2422, 23sylan 486 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2524iftrued 4039 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → if((𝐹𝑥) ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜)
2625mpteq2dva 4662 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1𝑜, ∅)) = (𝑥𝑋 ↦ 1𝑜))
2718ffvelrnda 6248 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2818feqmptd 6140 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
29 eqidd 2606 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)))
30 eleq1 2671 . . . . . . . . . . . . 13 (𝑎 = (𝐹𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
3130ifbid 4053 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = if((𝐹𝑥) ∈ ran 𝐹, 1𝑜, ∅))
3227, 28, 29, 31fmptco 6284 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) ∘ 𝐹) = (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1𝑜, ∅)))
33 fconstmpt 5071 . . . . . . . . . . . . 13 (𝑌 × {1𝑜}) = (𝑎𝑌 ↦ 1𝑜)
3433a1i 11 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1𝑜}) = (𝑎𝑌 ↦ 1𝑜))
35 eqidd 2606 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → 1𝑜 = 1𝑜)
3627, 28, 34, 35fmptco 6284 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1𝑜}) ∘ 𝐹) = (𝑥𝑋 ↦ 1𝑜))
3726, 32, 363eqtr4d 2649 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) ∘ 𝐹) = ((𝑌 × {1𝑜}) ∘ 𝐹))
385adantr 479 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈𝑉)
399adantr 479 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋𝑈)
4012adantr 479 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌𝑈)
41 setcepi.2 . . . . . . . . . . . 12 (𝜑 → 2𝑜𝑈)
4241adantr 479 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2𝑜𝑈)
43 eqid 2605 . . . . . . . . . . . . 13 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅))
44 1onn 7579 . . . . . . . . . . . . . . . . . 18 1𝑜 ∈ ω
4544elexi 3181 . . . . . . . . . . . . . . . . 17 1𝑜 ∈ V
4645prid2 4237 . . . . . . . . . . . . . . . 16 1𝑜 ∈ {∅, 1𝑜}
47 df2o3 7433 . . . . . . . . . . . . . . . 16 2𝑜 = {∅, 1𝑜}
4846, 47eleqtrri 2682 . . . . . . . . . . . . . . 15 1𝑜 ∈ 2𝑜
49 0ex 4709 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
5049prid1 4236 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅, 1𝑜}
5150, 47eleqtrri 2682 . . . . . . . . . . . . . . 15 ∅ ∈ 2𝑜
5248, 51keepel 4100 . . . . . . . . . . . . . 14 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) ∈ 2𝑜
5352a1i 11 . . . . . . . . . . . . 13 (𝑎𝑌 → if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) ∈ 2𝑜)
5443, 53fmpti 6272 . . . . . . . . . . . 12 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)):𝑌⟶2𝑜
5554a1i 11 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)):𝑌⟶2𝑜)
566, 38, 3, 39, 40, 42, 18, 55setcco 16498 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2𝑜)𝐹) = ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) ∘ 𝐹))
57 fconst6g 5988 . . . . . . . . . . . 12 (1𝑜 ∈ 2𝑜 → (𝑌 × {1𝑜}):𝑌⟶2𝑜)
5848, 57mp1i 13 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1𝑜}):𝑌⟶2𝑜)
596, 38, 3, 39, 40, 42, 18, 58setcco 16498 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1𝑜})(⟨𝑋, 𝑌⟩(comp‘𝐶)2𝑜)𝐹) = ((𝑌 × {1𝑜}) ∘ 𝐹))
6037, 56, 593eqtr4d 2649 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2𝑜)𝐹) = ((𝑌 × {1𝑜})(⟨𝑋, 𝑌⟩(comp‘𝐶)2𝑜)𝐹))
618adantr 479 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
6211adantr 479 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
6313adantr 479 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
6441, 10eleqtrd 2685 . . . . . . . . . . 11 (𝜑 → 2𝑜 ∈ (Base‘𝐶))
6564adantr 479 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2𝑜 ∈ (Base‘𝐶))
66 simpr 475 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
676, 38, 2, 40, 42elsetchom 16496 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) ∈ (𝑌(Hom ‘𝐶)2𝑜) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)):𝑌⟶2𝑜))
6855, 67mpbird 245 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) ∈ (𝑌(Hom ‘𝐶)2𝑜))
696, 38, 2, 40, 42elsetchom 16496 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1𝑜}) ∈ (𝑌(Hom ‘𝐶)2𝑜) ↔ (𝑌 × {1𝑜}):𝑌⟶2𝑜))
7058, 69mpbird 245 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1𝑜}) ∈ (𝑌(Hom ‘𝐶)2𝑜))
711, 2, 3, 4, 61, 62, 63, 65, 66, 68, 70epii 16168 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2𝑜)𝐹) = ((𝑌 × {1𝑜})(⟨𝑋, 𝑌⟩(comp‘𝐶)2𝑜)𝐹) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑌 × {1𝑜})))
7260, 71mpbid 220 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑌 × {1𝑜}))
7372, 33syl6eq 2655 . . . . . . 7 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑎𝑌 ↦ 1𝑜))
7452rgenw 2903 . . . . . . . 8 𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) ∈ 2𝑜
75 mpteqb 6188 . . . . . . . 8 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) ∈ 2𝑜 → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑎𝑌 ↦ 1𝑜) ↔ ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜))
7674, 75ax-mp 5 . . . . . . 7 ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅)) = (𝑎𝑌 ↦ 1𝑜) ↔ ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜)
7773, 76sylib 206 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜)
78 1n0 7435 . . . . . . . . . 10 1𝑜 ≠ ∅
7978nesymi 2834 . . . . . . . . 9 ¬ ∅ = 1𝑜
80 iffalse 4040 . . . . . . . . . 10 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = ∅)
8180eqeq1d 2607 . . . . . . . . 9 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜 ↔ ∅ = 1𝑜))
8279, 81mtbiri 315 . . . . . . . 8 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜)
8382con4i 111 . . . . . . 7 (if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜𝑎 ∈ ran 𝐹)
8483ralimi 2931 . . . . . 6 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1𝑜, ∅) = 1𝑜 → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8577, 84syl 17 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
86 dfss3 3553 . . . . 5 (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8785, 86sylibr 222 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
8820, 87eqssd 3580 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
89 dffo2 6013 . . 3 (𝐹:𝑋onto𝑌 ↔ (𝐹:𝑋𝑌 ∧ ran 𝐹 = 𝑌))
9018, 88, 89sylanbrc 694 . 2 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋onto𝑌)
91 fof 6009 . . . . 5 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
9291adantl 480 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → 𝐹:𝑋𝑌)
9316biimpar 500 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9492, 93syldan 485 . . 3 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9510adantr 479 . . . . . 6 ((𝜑𝐹:𝑋onto𝑌) → 𝑈 = (Base‘𝐶))
9695eleq2d 2668 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
975ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
989ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋𝑈)
9912ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌𝑈)
100 simprl 789 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
10192adantr 479 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋𝑌)
102 simprrl 799 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
1036, 97, 2, 99, 100elsetchom 16496 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌𝑧))
104102, 103mpbid 220 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌𝑧)
1056, 97, 3, 98, 99, 100, 101, 104setcco 16498 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔𝐹))
106 simprrr 800 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
1076, 97, 2, 99, 100elsetchom 16496 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ :𝑌𝑧))
108106, 107mpbid 220 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → :𝑌𝑧)
1096, 97, 3, 98, 99, 100, 101, 108setcco 16498 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝐹))
110105, 109eqeq12d 2620 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) ↔ (𝑔𝐹) = (𝐹)))
111 simplr 787 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋onto𝑌)
112 ffn 5940 . . . . . . . . . . . 12 (𝑔:𝑌𝑧𝑔 Fn 𝑌)
113104, 112syl 17 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
114 ffn 5940 . . . . . . . . . . . 12 (:𝑌𝑧 Fn 𝑌)
115108, 114syl 17 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → Fn 𝑌)
116 cocan2 6421 . . . . . . . . . . 11 ((𝐹:𝑋onto𝑌𝑔 Fn 𝑌 Fn 𝑌) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
117111, 113, 115, 116syl3anc 1317 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
118117biimpd 217 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) → 𝑔 = ))
119110, 118sylbid 228 . . . . . . . 8 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
120119anassrs 677 . . . . . . 7 ((((𝜑𝐹:𝑋onto𝑌) ∧ 𝑧𝑈) ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
121120ralrimivva 2949 . . . . . 6 (((𝜑𝐹:𝑋onto𝑌) ∧ 𝑧𝑈) → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
122121ex 448 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈 → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = )))
12396, 122sylbird 248 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → (𝑧 ∈ (Base‘𝐶) → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = )))
124123ralrimiv 2943 . . 3 ((𝜑𝐹:𝑋onto𝑌) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
1251, 2, 3, 4, 8, 11, 13isepi2 16166 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))))
126125adantr 479 . . 3 ((𝜑𝐹:𝑋onto𝑌) → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))))
12794, 124, 126mpbir2and 958 . 2 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
12890, 127impbida 872 1 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wral 2891  wss 3535  c0 3869  ifcif 4031  {csn 4120  {cpr 4122  cop 4126  cmpt 4633   × cxp 5022  ran crn 5025  ccom 5028   Fn wfn 5781  wf 5782  ontowfo 5784  cfv 5786  (class class class)co 6523  ωcom 6930  1𝑜c1o 7413  2𝑜c2o 7414  Basecbs 15637  Hom chom 15721  compcco 15722  Catccat 16090  Epicepi 16154  SetCatcsetc 16490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-tpos 7212  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-fz 12149  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-hom 15735  df-cco 15736  df-cat 16094  df-cid 16095  df-oppc 16137  df-mon 16155  df-epi 16156  df-setc 16491
This theorem is referenced by: (None)
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