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Theorem rexrnmpt2 5643
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
rexrnmpt2 (∀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 rexrnmpt2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpt2 5638 . . . 4 ran 𝐹 = {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}
32rexeqi 2527 . . 3 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑)
4 eqeq1 2062 . . . . 5 (𝑤 = 𝑧 → (𝑤 = 𝐶𝑧 = 𝐶))
542rexbidv 2366 . . . 4 (𝑤 = 𝑧 → (∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶))
65rexab 2725 . . 3 (∃𝑧 ∈ {𝑤 ∣ ∃𝑥𝐴𝑦𝐵 𝑤 = 𝐶}𝜑 ↔ ∃𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
7 rexcom4 2594 . . . 4 (∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑))
8 r19.41v 2483 . . . . 5 (∃𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
98exbii 1512 . . . 4 (∃𝑧𝑥𝐴 (∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑))
107, 9bitr2i 178 . . 3 (∃𝑧(∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
113, 6, 103bitri 199 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
12 rexcom4 2594 . . . . . 6 (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑧𝑦𝐵 (𝑧 = 𝐶𝜑))
13 r19.41v 2483 . . . . . . 7 (∃𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ (∃𝑦𝐵 𝑧 = 𝐶𝜑))
1413exbii 1512 . . . . . 6 (∃𝑧𝑦𝐵 (𝑧 = 𝐶𝜑) ↔ ∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
1512, 14bitri 177 . . . . 5 (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑))
16 ralrnmpt2.2 . . . . . . . 8 (𝑧 = 𝐶 → (𝜑𝜓))
1716ceqsexgv 2695 . . . . . . 7 (𝐶𝑉 → (∃𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
1817ralimi 2401 . . . . . 6 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (∃𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓))
19 rexbi 2463 . . . . . 6 (∀𝑦𝐵 (∃𝑧(𝑧 = 𝐶𝜑) ↔ 𝜓) → (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
2018, 19syl 14 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → (∃𝑦𝐵𝑧(𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
2115, 20syl5bbr 187 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
2221ralimi 2401 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∃𝑧(∃𝑦𝐵 𝑧 = 𝐶𝜑) ↔ ∃𝑦𝐵 𝜓))
23 rexbi 2463 . . 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, 24syl5bb 185 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wral 2323  wrex 2324  ran crn 4373  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-cnv 4380  df-dm 4382  df-rn 4383  df-oprab 5543  df-mpt2 5544
This theorem is referenced by: (None)
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