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Theorem foco2OLD 6288
Description: Obsolete proof of foco2 6287 as of 14-Jul-2021. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
foco2OLD ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)

Proof of Theorem foco2OLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1054 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵𝐶)
2 foelrn 6286 . . . . . 6 (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧))
3 ffvelrn 6265 . . . . . . . . . 10 ((𝐺:𝐴𝐵𝑧𝐴) → (𝐺𝑧) ∈ 𝐵)
43adantll 746 . . . . . . . . 9 (((𝐹:𝐵𝐶𝐺:𝐴𝐵) ∧ 𝑧𝐴) → (𝐺𝑧) ∈ 𝐵)
5 fvco3 6185 . . . . . . . . . 10 ((𝐺:𝐴𝐵𝑧𝐴) → ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧)))
65adantll 746 . . . . . . . . 9 (((𝐹:𝐵𝐶𝐺:𝐴𝐵) ∧ 𝑧𝐴) → ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧)))
7 fveq2 6103 . . . . . . . . . . 11 (𝑥 = (𝐺𝑧) → (𝐹𝑥) = (𝐹‘(𝐺𝑧)))
87eqeq2d 2620 . . . . . . . . . 10 (𝑥 = (𝐺𝑧) → (((𝐹𝐺)‘𝑧) = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))))
98rspcev 3282 . . . . . . . . 9 (((𝐺𝑧) ∈ 𝐵 ∧ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
104, 6, 9syl2anc 691 . . . . . . . 8 (((𝐹:𝐵𝐶𝐺:𝐴𝐵) ∧ 𝑧𝐴) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
11 eqeq1 2614 . . . . . . . . 9 (𝑦 = ((𝐹𝐺)‘𝑧) → (𝑦 = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
1211rexbidv 3034 . . . . . . . 8 (𝑦 = ((𝐹𝐺)‘𝑧) → (∃𝑥𝐵 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
1310, 12syl5ibrcom 236 . . . . . . 7 (((𝐹:𝐵𝐶𝐺:𝐴𝐵) ∧ 𝑧𝐴) → (𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1413rexlimdva 3013 . . . . . 6 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
152, 14syl5 33 . . . . 5 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1615impl 648 . . . 4 ((((𝐹:𝐵𝐶𝐺:𝐴𝐵) ∧ (𝐹𝐺):𝐴onto𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
1716ralrimiva 2949 . . 3 (((𝐹:𝐵𝐶𝐺:𝐴𝐵) ∧ (𝐹𝐺):𝐴onto𝐶) → ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥))
18173impa 1251 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥))
19 dffo3 6282 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
201, 18, 19sylanbrc 695 1 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  ccom 5042  wf 5800  ontowfo 5802  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812
This theorem is referenced by: (None)
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