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Theorem ralrnmpt2 5615
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 5611 . . . 4 ran 𝐹 = {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}
32raleqi 2509 . . 3 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑)
4 eqeq1 2046 . . . . 5 (𝑤 = 𝑧 → (𝑤 = 𝐶𝑧 = 𝐶))
542rexbidv 2349 . . . 4 (𝑤 = 𝑧 → (∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶))
65ralab 2701 . . 3 (∀𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
7 ralcom4 2576 . . . 4 (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑))
8 r19.23v 2425 . . . . 5 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
98albii 1359 . . . 4 (∀𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
107, 9bitr2i 174 . . 3 (∀𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
113, 6, 103bitri 195 . 2 (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
12 ralcom4 2576 . . . . . 6 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑))
13 r19.23v 2425 . . . . . . 7 (∀𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ (∃𝑦𝐵 𝑧 = 𝐶𝜑))
1413albii 1359 . . . . . 6 (∀𝑧𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
1512, 14bitri 173 . . . . 5 (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
16 nfv 1421 . . . . . . . 8 𝑧𝜓
17 ralrnmpt2.2 . . . . . . . 8 (𝑧 = 𝐶 → (𝜑𝜓))
1816, 17ceqsalg 2582 . . . . . . 7 (𝐶𝑉 → (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
1918ralimi 2384 . . . . . 6 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
20 ralbi 2445 . . . . . 6 (∀𝑦𝐵 (∀𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓) → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2119, 20syl 14 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2215, 21syl5bbr 183 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
2322ralimi 2384 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓))
24 ralbi 2445 . . 3 (∀𝑥𝐴 (∀𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑦𝐵 𝜓) → (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
2523, 24syl 14 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
2611, 25syl5bb 181 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  ran crn 4346  cmpt2 5514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-oprab 5516  df-mpt2 5517
This theorem is referenced by: (None)
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