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Theorem ralrnmpo 7557
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
ralrnmpo.2 (𝑧 = 𝐶 → (𝜑𝜓))
Assertion
Ref Expression
ralrnmpo (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝐹   𝜓,𝑧   𝑥,𝑦,𝑧   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem ralrnmpo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpo 7551 . . . 4 ran 𝐹 = {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}
32raleqi 3313 . . 3 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑)
4 eqeq1 2729 . . . . 5 (𝑤 = 𝑧 → (𝑤 = 𝐶𝑧 = 𝐶))
542rexbidv 3210 . . . 4 (𝑤 = 𝑧 → (∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶))
65ralab 3678 . . 3 (∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
7 ralcom4 3274 . . . 4 (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑))
8 r19.23v 3173 . . . . 5 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
98albii 1813 . . . 4 (∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
107, 9bitr2i 275 . . 3 (∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
113, 6, 103bitri 296 . 2 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
12 ralcom4 3274 . . . . . 6 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑))
13 r19.23v 3173 . . . . . . 7 (∀𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ (∃𝑦𝐵 𝑧 = 𝐶𝜑))
1413albii 1813 . . . . . 6 (∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
1512, 14bitri 274 . . . . 5 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
16 nfv 1909 . . . . . . . 8 𝑧𝜓
17 ralrnmpo.2 . . . . . . . 8 (𝑧 = 𝐶 → (𝜑𝜓))
1816, 17ceqsalg 3497 . . . . . . 7 (𝐶𝑉 → (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
1918ralimi 3073 . . . . . 6 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
20 ralbi 3093 . . . . . 6 (∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓) → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2119, 20syl 17 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2215, 21bitr3id 284 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2322ralimi 3073 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
24 ralbi 3093 . . 3 (∀𝑥𝐴 (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓) → (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
2523, 24syl 17 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
2611, 25bitrid 282 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wcel 2098  {cab 2702  wral 3051  wrex 3060  ran crn 5673  cmpo 7418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-oprab 7420  df-mpo 7421
This theorem is referenced by:  rexrnmpo  7558  efgval2  19683  txcnp  23542  txcnmpt  23546  txflf  23928
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