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Theorem ralrnmpo 7278
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 7273 . . . 4 ran 𝐹 = {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}
32raleqi 3401 . . 3 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑)
4 eqeq1 2828 . . . . 5 (𝑤 = 𝑧 → (𝑤 = 𝐶𝑧 = 𝐶))
542rexbidv 3293 . . . 4 (𝑤 = 𝑧 → (∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶))
65ralab 3670 . . 3 (∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
7 ralcom4 3230 . . . 4 (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑))
8 r19.23v 3272 . . . . 5 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
98albii 1821 . . . 4 (∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
107, 9bitr2i 279 . . 3 (∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
113, 6, 103bitri 300 . 2 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
12 ralcom4 3230 . . . . . 6 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑))
13 r19.23v 3272 . . . . . . 7 (∀𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ (∃𝑦𝐵 𝑧 = 𝐶𝜑))
1413albii 1821 . . . . . 6 (∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
1512, 14bitri 278 . . . . 5 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
16 nfv 1916 . . . . . . . 8 𝑧𝜓
17 ralrnmpo.2 . . . . . . . 8 (𝑧 = 𝐶 → (𝜑𝜓))
1816, 17ceqsalg 3515 . . . . . . 7 (𝐶𝑉 → (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
1918ralimi 3155 . . . . . 6 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
20 ralbi 3162 . . . . . 6 (∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓) → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2119, 20syl 17 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2215, 21bitr3id 288 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2322ralimi 3155 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
24 ralbi 3162 . . 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, 25syl5bb 286 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  ran crn 5543  cmpo 7147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-cnv 5550  df-dm 5552  df-rn 5553  df-oprab 7149  df-mpo 7150
This theorem is referenced by:  rexrnmpo  7279  efgval2  18846  txcnp  22221  txcnmpt  22225  txflf  22607
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