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Theorem foeqcnvco 7320
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 6875 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
2 cnveq 5887 . . . . . 6 (𝐹 = 𝐺𝐹 = 𝐺)
32coeq2d 5876 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
43eqeq1d 2737 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐵) ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
51, 4syl5ibcom 245 . . 3 (𝐹:𝐴onto𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
65adantr 480 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
7 fofn 6823 . . . . 5 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
87ad2antrr 726 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9 fofn 6823 . . . . 5 (𝐺:𝐴onto𝐵𝐺 Fn 𝐴)
109ad2antlr 727 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
119adantl 481 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺 Fn 𝐴)
12 fnopfv 7095 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1311, 12sylan 580 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
14 fvex 6920 . . . . . . . . . . . . 13 (𝐺𝑥) ∈ V
15 vex 3482 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15brcnv 5896 . . . . . . . . . . . 12 ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥))
17 df-br 5149 . . . . . . . . . . . 12 (𝑥𝐺(𝐺𝑥) ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1816, 17bitri 275 . . . . . . . . . . 11 ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1913, 18sylibr 234 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)𝐺𝑥)
207adantr 480 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹 Fn 𝐴)
21 fnopfv 7095 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2220, 21sylan 580 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
23 df-br 5149 . . . . . . . . . . 11 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2422, 23sylibr 234 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
25 breq2 5152 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐺𝑥)𝐺𝑦 ↔ (𝐺𝑥)𝐺𝑥))
26 breq1 5151 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝐹(𝐹𝑥) ↔ 𝑥𝐹(𝐹𝑥)))
2725, 26anbi12d 632 . . . . . . . . . . 11 (𝑦 = 𝑥 → (((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)) ↔ ((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥))))
2815, 27spcev 3606 . . . . . . . . . 10 (((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥)) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
2919, 24, 28syl2anc 584 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
30 fvex 6920 . . . . . . . . . 10 (𝐹𝑥) ∈ V
3114, 30brco 5884 . . . . . . . . 9 ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3229, 31sylibr 234 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
3332adantlr 715 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
34 breq 5150 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐵) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
3534ad2antlr 727 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
3633, 35mpbid 232 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥))
37 fof 6821 . . . . . . . . . 10 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
3837adantl 481 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺:𝐴𝐵)
3938ffvelcdmda 7104 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
40 fof 6821 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
4140adantr 480 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹:𝐴𝐵)
4241ffvelcdmda 7104 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
43 resieq 6011 . . . . . . . 8 (((𝐺𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4439, 42, 43syl2anc 584 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4544adantlr 715 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
4636, 45mpbid 232 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
4746eqcomd 2741 . . . 4 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
488, 10, 47eqfnfvd 7054 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
4948ex 412 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → ((𝐹𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
506, 49impbid 212 1 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  cop 4637   class class class wbr 5148   I cid 5582  ccnv 5688  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by: (None)
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