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Theorem dffo3f 39863
Description: An onto mapping expressed in terms of function values. As dffo3 6537 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 6280 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 6206 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 6404 . . . . . . 7 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)})
4 nfcv 2902 . . . . . . . . . . 11 𝑥𝑦
5 dffo3f.1 . . . . . . . . . . . 12 𝑥𝐹
6 nfcv 2902 . . . . . . . . . . . 12 𝑥𝑤
75, 6nffv 6359 . . . . . . . . . . 11 𝑥(𝐹𝑤)
84, 7nfeq 2914 . . . . . . . . . 10 𝑥 𝑦 = (𝐹𝑤)
9 nfv 1992 . . . . . . . . . 10 𝑤 𝑦 = (𝐹𝑥)
10 fveq2 6352 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
1110eqeq2d 2770 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑦 = (𝐹𝑤) ↔ 𝑦 = (𝐹𝑥)))
128, 9, 11cbvrex 3307 . . . . . . . . 9 (∃𝑤𝐴 𝑦 = (𝐹𝑤) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1312abbii 2877 . . . . . . . 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 2794 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1615eqeq1d 2762 . . . . 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 2902 . . . . . . . . . 10 𝑥𝐴
19 nfcv 2902 . . . . . . . . . 10 𝑥𝐵
205, 18, 19nff 6202 . . . . . . . . 9 𝑥 𝐹:𝐴𝐵
21 nfv 1992 . . . . . . . . 9 𝑥 𝑦𝐵
22 simpr 479 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
23 ffvelrn 6520 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
2423adantr 472 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
2522, 24eqeltrd 2839 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
2625exp31 631 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦𝐵)))
2720, 21, 26rexlimd 3164 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
2827biantrurd 530 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
29 dfbi2 663 . . . . . . 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 1998 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
32 abeq1 2871 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
33 df-ral 3055 . . . . 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 672 . 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 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wnfc 2889  wral 3050  wrex 3051  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057
This theorem is referenced by:  foelrnf  39872  fompt  39878
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