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Theorem foeqcnvco 5607
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 5314 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
2 cnveq 4641 . . . . . 6 (𝐹 = 𝐺𝐹 = 𝐺)
32coeq2d 4629 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
43eqeq1d 2103 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐵) ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
51, 4syl5ibcom 154 . . 3 (𝐹:𝐴onto𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
65adantr 271 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
7 fofn 5270 . . . . 5 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
87ad2antrr 473 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9 fofn 5270 . . . . 5 (𝐺:𝐴onto𝐵𝐺 Fn 𝐴)
109ad2antlr 474 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
119adantl 272 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺 Fn 𝐴)
12 fnopfv 5468 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1311, 12sylan 278 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
149anim1i 334 . . . . . . . . . . . . 13 ((𝐺:𝐴onto𝐵𝑥𝐴) → (𝐺 Fn 𝐴𝑥𝐴))
1514adantll 461 . . . . . . . . . . . 12 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺 Fn 𝐴𝑥𝐴))
16 funfvex 5357 . . . . . . . . . . . . . . 15 ((Fun 𝐺𝑥 ∈ dom 𝐺) → (𝐺𝑥) ∈ V)
1716funfni 5148 . . . . . . . . . . . . . 14 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ V)
18 vex 2636 . . . . . . . . . . . . . 14 𝑥 ∈ V
19 brcnvg 4648 . . . . . . . . . . . . . 14 (((𝐺𝑥) ∈ V ∧ 𝑥 ∈ V) → ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥)))
2017, 18, 19sylancl 405 . . . . . . . . . . . . 13 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥)))
21 df-br 3868 . . . . . . . . . . . . 13 (𝑥𝐺(𝐺𝑥) ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
2220, 21syl6bb 195 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺))
2315, 22syl 14 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺))
2413, 23mpbird 166 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)𝐺𝑥)
257adantr 271 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹 Fn 𝐴)
26 fnopfv 5468 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2725, 26sylan 278 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
28 df-br 3868 . . . . . . . . . . 11 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2927, 28sylibr 133 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
30 breq2 3871 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐺𝑥)𝐺𝑦 ↔ (𝐺𝑥)𝐺𝑥))
31 breq1 3870 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝐹(𝐹𝑥) ↔ 𝑥𝐹(𝐹𝑥)))
3230, 31anbi12d 458 . . . . . . . . . . 11 (𝑦 = 𝑥 → (((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)) ↔ ((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥))))
3318, 32spcev 2727 . . . . . . . . . 10 (((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥)) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3424, 29, 33syl2anc 404 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3515, 17syl 14 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ V)
367anim1i 334 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
3736adantlr 462 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
38 funfvex 5357 . . . . . . . . . . . 12 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
3938funfni 5148 . . . . . . . . . . 11 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
4037, 39syl 14 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ V)
41 brcog 4634 . . . . . . . . . 10 (((𝐺𝑥) ∈ V ∧ (𝐹𝑥) ∈ V) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥))))
4235, 40, 41syl2anc 404 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥))))
4334, 42mpbird 166 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
4443adantlr 462 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
45 breq 3869 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐵) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
4645ad2antlr 474 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
4744, 46mpbid 146 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥))
48 fof 5268 . . . . . . . . . 10 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
4948adantl 272 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺:𝐴𝐵)
5049ffvelrnda 5473 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
51 fof 5268 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5251adantr 271 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹:𝐴𝐵)
5352ffvelrnda 5473 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
54 resieq 4755 . . . . . . . 8 (((𝐺𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
5550, 53, 54syl2anc 404 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
5655adantlr 462 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
5747, 56mpbid 146 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
5857eqcomd 2100 . . . 4 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
598, 10, 58eqfnfvd 5439 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
6059ex 114 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → ((𝐹𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
616, 60impbid 128 1 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wex 1433  wcel 1445  Vcvv 2633  cop 3469   class class class wbr 3867   I cid 4139  ccnv 4466  cres 4469  ccom 4471   Fn wfn 5044  wf 5045  ontowfo 5047  cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057
This theorem is referenced by: (None)
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