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Theorem dffo3f 38869
Description: An onto mapping expressed in terms of function values. As dffo3 6335 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
dffo3f.1 𝑥𝐹
Assertion
Ref Expression
dffo3f (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐹
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dffo3f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dffo2 6081 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 6007 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 6204 . . . . . . 7 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)})
4 nfcv 2761 . . . . . . . . . . 11 𝑥𝑦
5 dffo3f.1 . . . . . . . . . . . 12 𝑥𝐹
6 nfcv 2761 . . . . . . . . . . . 12 𝑥𝑤
75, 6nffv 6160 . . . . . . . . . . 11 𝑥(𝐹𝑤)
84, 7nfeq 2772 . . . . . . . . . 10 𝑥 𝑦 = (𝐹𝑤)
9 nfv 1840 . . . . . . . . . 10 𝑤 𝑦 = (𝐹𝑥)
10 fveq2 6153 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
1110eqeq2d 2631 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑦 = (𝐹𝑤) ↔ 𝑦 = (𝐹𝑥)))
128, 9, 11cbvrex 3159 . . . . . . . . 9 (∃𝑤𝐴 𝑦 = (𝐹𝑤) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1312abbii 2736 . . . . . . . 8 {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
1413a1i 11 . . . . . . 7 (𝐹 Fn 𝐴 → {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
153, 14eqtrd 2655 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615eqeq1d 2623 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
172, 16syl 17 . . . 4 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
18 nfcv 2761 . . . . . . . . . 10 𝑥𝐴
19 nfcv 2761 . . . . . . . . . 10 𝑥𝐵
205, 18, 19nff 6003 . . . . . . . . 9 𝑥 𝐹:𝐴𝐵
21 nfv 1840 . . . . . . . . 9 𝑥 𝑦𝐵
22 simpr 477 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
23 ffvelrn 6318 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
2423adantr 481 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
2522, 24eqeltrd 2698 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
2625exp31 629 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦𝐵)))
2720, 21, 26rexlimd 3020 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
2827biantrurd 529 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
29 dfbi2 659 . . . . . . 7 ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
3028, 29syl6rbbr 279 . . . . . 6 (𝐹:𝐴𝐵 → ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
3130albidv 1846 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
32 abeq1 2730 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
33 df-ral 2912 . . . . 5 (∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
3431, 32, 333bitr4g 303 . . . 4 (𝐹:𝐴𝐵 → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3517, 34bitrd 268 . . 3 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3635pm5.32i 668 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
371, 36bitri 264 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wnfc 2748  wral 2907  wrex 2908  ran crn 5080   Fn wfn 5847  wf 5848  ontowfo 5850  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860
This theorem is referenced by:  foelrnf  38878  fompt  38884
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