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Theorem foeqcnvco 7059
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 6643 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
2 cnveq 5747 . . . . . 6 (𝐹 = 𝐺𝐹 = 𝐺)
32coeq2d 5736 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
43eqeq1d 2826 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐵) ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
51, 4syl5ibcom 247 . . 3 (𝐹:𝐴onto𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
65adantr 483 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
7 fofn 6595 . . . . 5 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
87ad2antrr 724 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9 fofn 6595 . . . . 5 (𝐺:𝐴onto𝐵𝐺 Fn 𝐴)
109ad2antlr 725 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
119adantl 484 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺 Fn 𝐴)
12 fnopfv 6846 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1311, 12sylan 582 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
14 fvex 6686 . . . . . . . . . . . . 13 (𝐺𝑥) ∈ V
15 vex 3500 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15brcnv 5756 . . . . . . . . . . . 12 ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥))
17 df-br 5070 . . . . . . . . . . . 12 (𝑥𝐺(𝐺𝑥) ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1816, 17bitri 277 . . . . . . . . . . 11 ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1913, 18sylibr 236 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)𝐺𝑥)
207adantr 483 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹 Fn 𝐴)
21 fnopfv 6846 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2220, 21sylan 582 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
23 df-br 5070 . . . . . . . . . . 11 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2422, 23sylibr 236 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
25 breq2 5073 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐺𝑥)𝐺𝑦 ↔ (𝐺𝑥)𝐺𝑥))
26 breq1 5072 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝐹(𝐹𝑥) ↔ 𝑥𝐹(𝐹𝑥)))
2725, 26anbi12d 632 . . . . . . . . . . 11 (𝑦 = 𝑥 → (((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)) ↔ ((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥))))
2815, 27spcev 3610 . . . . . . . . . 10 (((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥)) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
2919, 24, 28syl2anc 586 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
30 fvex 6686 . . . . . . . . . 10 (𝐹𝑥) ∈ V
3114, 30brco 5744 . . . . . . . . 9 ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3229, 31sylibr 236 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
3332adantlr 713 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
34 breq 5071 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐵) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
3534ad2antlr 725 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
3633, 35mpbid 234 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥))
37 fof 6593 . . . . . . . . . 10 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
3837adantl 484 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺:𝐴𝐵)
3938ffvelrnda 6854 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
40 fof 6593 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
4140adantr 483 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹:𝐴𝐵)
4241ffvelrnda 6854 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
43 resieq 5867 . . . . . . . 8 (((𝐺𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4439, 42, 43syl2anc 586 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4544adantlr 713 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4636, 45mpbid 234 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
4746eqcomd 2830 . . . 4 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
488, 10, 47eqfnfvd 6808 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
4948ex 415 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → ((𝐹𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
506, 49impbid 214 1 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wex 1779  wcel 2113  cop 4576   class class class wbr 5069   I cid 5462  ccnv 5557  cres 5560  ccom 5562   Fn wfn 6353  wf 6354  ontowfo 6356  cfv 6358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366
This theorem is referenced by: (None)
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