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Theorem rexrnmpo 5989
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
rexrnmpo (∀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 rexrnmpo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpo 5984 . . . 4 ran 𝐹 = {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}
32rexeqi 2677 . . 3 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑)
4 eqeq1 2184 . . . . 5 (𝑤 = 𝑧 → (𝑤 = 𝐶𝑧 = 𝐶))
542rexbidv 2502 . . . 4 (𝑤 = 𝑧 → (∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶))
65rexab 2899 . . 3 (∃𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑 ↔ ∃𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
7 rexcom4 2760 . . . 4 (∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑))
8 r19.41v 2633 . . . . 5 (∃𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
98exbii 1605 . . . 4 (∃𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
107, 9bitr2i 185 . . 3 (∃𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
113, 6, 103bitri 206 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
12 rexcom4 2760 . . . . . 6 (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑧𝑦𝐵 (𝑧 = 𝐶𝜑))
13 r19.41v 2633 . . . . . . 7 (∃𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ (∃𝑦𝐵 𝑧 = 𝐶𝜑))
1413exbii 1605 . . . . . 6 (∃𝑧𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ ∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
1512, 14bitri 184 . . . . 5 (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
16 ralrnmpo.2 . . . . . . . 8 (𝑧 = 𝐶 → (𝜑𝜓))
1716ceqsexgv 2866 . . . . . . 7 (𝐶𝑉 → (∃𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
1817ralimi 2540 . . . . . 6 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (∃𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
19 rexbi 2610 . . . . . 6 (∀𝑦𝐵 (∃𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓) → (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
2018, 19syl 14 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
2115, 20bitr3id 194 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
2221ralimi 2540 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
23 rexbi 2610 . . 3 (∀𝑥𝐴 (∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓) → (∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
2422, 23syl 14 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
2511, 24bitrid 192 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  ran crn 4627  cmpo 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-cnv 4634  df-dm 4636  df-rn 4637  df-oprab 5878  df-mpo 5879
This theorem is referenced by:  eltx  13652  txrest  13669  txlm  13672
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