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Theorem ralrnmpt2 6817
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
ralrnmpt2.2 (𝑧 = 𝐶 → (𝜑𝜓))
Assertion
Ref Expression
ralrnmpt2 (∀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 ralrnmpt2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpt2 6812 . . . 4 ran 𝐹 = {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}
32raleqi 3172 . . 3 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑)
4 eqeq1 2655 . . . . 5 (𝑤 = 𝑧 → (𝑤 = 𝐶𝑧 = 𝐶))
542rexbidv 3086 . . . 4 (𝑤 = 𝑧 → (∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶))
65ralab 3400 . . 3 (∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
7 ralcom4 3255 . . . 4 (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑))
8 r19.23v 3052 . . . . 5 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
98albii 1787 . . . 4 (∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
107, 9bitr2i 265 . . 3 (∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
113, 6, 103bitri 286 . 2 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
12 ralcom4 3255 . . . . . 6 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑))
13 r19.23v 3052 . . . . . . 7 (∀𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ (∃𝑦𝐵 𝑧 = 𝐶𝜑))
1413albii 1787 . . . . . 6 (∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
1512, 14bitri 264 . . . . 5 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
16 nfv 1883 . . . . . . . 8 𝑧𝜓
17 ralrnmpt2.2 . . . . . . . 8 (𝑧 = 𝐶 → (𝜑𝜓))
1816, 17ceqsalg 3261 . . . . . . 7 (𝐶𝑉 → (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
1918ralimi 2981 . . . . . 6 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
20 ralbi 3097 . . . . . 6 (∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓) → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2119, 20syl 17 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2215, 21syl5bbr 274 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2322ralimi 2981 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
24 ralbi 3097 . . 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 272 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  ran crn 5144  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  rexrnmpt2  6818  efgval2  18183  txcnp  21471  txcnmpt  21475  txflf  21857
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