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Theorem foeqcnvco 6540
Description: Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
Assertion
Ref Expression
foeqcnvco ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))

Proof of Theorem foeqcnvco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fococnv2 6149 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
2 cnveq 5285 . . . . . 6 (𝐹 = 𝐺𝐹 = 𝐺)
32coeq2d 5273 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
43eqeq1d 2622 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐵) ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
51, 4syl5ibcom 235 . . 3 (𝐹:𝐴onto𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
65adantr 481 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
7 fofn 6104 . . . . 5 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
87ad2antrr 761 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9 fofn 6104 . . . . 5 (𝐺:𝐴onto𝐵𝐺 Fn 𝐴)
109ad2antlr 762 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
119adantl 482 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺 Fn 𝐴)
12 fnopfv 6337 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1311, 12sylan 488 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
14 fvex 6188 . . . . . . . . . . . . 13 (𝐺𝑥) ∈ V
15 vex 3198 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15brcnv 5294 . . . . . . . . . . . 12 ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥))
17 df-br 4645 . . . . . . . . . . . 12 (𝑥𝐺(𝐺𝑥) ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1816, 17bitri 264 . . . . . . . . . . 11 ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1913, 18sylibr 224 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)𝐺𝑥)
207adantr 481 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹 Fn 𝐴)
21 fnopfv 6337 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2220, 21sylan 488 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
23 df-br 4645 . . . . . . . . . . 11 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2422, 23sylibr 224 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
25 breq2 4648 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐺𝑥)𝐺𝑦 ↔ (𝐺𝑥)𝐺𝑥))
26 breq1 4647 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝐹(𝐹𝑥) ↔ 𝑥𝐹(𝐹𝑥)))
2725, 26anbi12d 746 . . . . . . . . . . 11 (𝑦 = 𝑥 → (((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)) ↔ ((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥))))
2815, 27spcev 3295 . . . . . . . . . 10 (((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥)) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
2919, 24, 28syl2anc 692 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
30 fvex 6188 . . . . . . . . . 10 (𝐹𝑥) ∈ V
3114, 30brco 5281 . . . . . . . . 9 ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3229, 31sylibr 224 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
3332adantlr 750 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
34 breq 4646 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐵) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
3534ad2antlr 762 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
3633, 35mpbid 222 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥))
37 fof 6102 . . . . . . . . . 10 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
3837adantl 482 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺:𝐴𝐵)
3938ffvelrnda 6345 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
40 fof 6102 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
4140adantr 481 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹:𝐴𝐵)
4241ffvelrnda 6345 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
43 resieq 5395 . . . . . . . 8 (((𝐺𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4439, 42, 43syl2anc 692 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4544adantlr 750 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4636, 45mpbid 222 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
4746eqcomd 2626 . . . 4 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
488, 10, 47eqfnfvd 6300 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
4948ex 450 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → ((𝐹𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
506, 49impbid 202 1 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  cop 4174   class class class wbr 4644   I cid 5013  ccnv 5103  cres 5106  ccom 5108   Fn wfn 5871  wf 5872  ontowfo 5874  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884
This theorem is referenced by: (None)
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