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Theorem dffo3f 7126
Description: An onto mapping expressed in terms of function values. As dffo3 7122 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 6824 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 6736 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 6968 . . . . . . 7 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)})
4 dffo3f.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2905 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6916 . . . . . . . . . 10 𝑥(𝐹𝑤)
76nfeq2 2923 . . . . . . . . 9 𝑥 𝑦 = (𝐹𝑤)
8 nfv 1914 . . . . . . . . 9 𝑤 𝑦 = (𝐹𝑥)
9 fveq2 6906 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
109eqeq2d 2748 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦 = (𝐹𝑤) ↔ 𝑦 = (𝐹𝑥)))
117, 8, 10cbvrexw 3307 . . . . . . . 8 (∃𝑤𝐴 𝑦 = (𝐹𝑤) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1211abbii 2809 . . . . . . 7 {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
133, 12eqtrdi 2793 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1413eqeq1d 2739 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
152, 14syl 17 . . . 4 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
16 dfbi2 474 . . . . . . 7 ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
17 nfcv 2905 . . . . . . . . . 10 𝑥𝐴
18 nfcv 2905 . . . . . . . . . 10 𝑥𝐵
194, 17, 18nff 6732 . . . . . . . . 9 𝑥 𝐹:𝐴𝐵
20 nfv 1914 . . . . . . . . 9 𝑥 𝑦𝐵
21 simpr 484 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
22 ffvelcdm 7101 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
2322adantr 480 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
2421, 23eqeltrd 2841 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
2524exp31 419 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦𝐵)))
2619, 20, 25rexlimd 3266 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
2726biantrurd 532 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
2816, 27bitr4id 290 . . . . . 6 (𝐹:𝐴𝐵 → ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
2928albidv 1920 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
30 eqabcb 2883 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
31 df-ral 3062 . . . . 5 (∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
3229, 30, 313bitr4g 314 . . . 4 (𝐹:𝐴𝐵 → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3315, 32bitrd 279 . . 3 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3433pm5.32i 574 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
351, 34bitri 275 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wnfc 2890  wral 3061  wrex 3070  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569
This theorem is referenced by:  foelrnf  7128  fompt  7138
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