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Theorem imaeqexov 7671
Description: Substitute an operation value into an existential quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.)
Hypothesis
Ref Expression
imaeqexov.1 (𝑥 = (𝑦𝐹𝑧) → (𝜑𝜓))
Assertion
Ref Expression
imaeqexov ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦𝐵𝑧𝐶 𝜓))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem imaeqexov
StepHypRef Expression
1 df-rex 3069 . 2 (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑))
2 ovelimab 7611 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧)))
32anbi1d 631 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ (∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
4 r19.41v 3187 . . . . . . 7 (∃𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ (∃𝑧𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
54rexbii 3092 . . . . . 6 (∃𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦𝐵 (∃𝑧𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
6 r19.41v 3187 . . . . . 6 (∃𝑦𝐵 (∃𝑧𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ (∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
75, 6bitr2i 276 . . . . 5 ((∃𝑦𝐵𝑧𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
83, 7bitrdi 287 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
98exbidv 1919 . . 3 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑥𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
10 rexcom4 3286 . . . 4 (∃𝑦𝐵𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
11 rexcom4 3286 . . . . . 6 (∃𝑧𝐶𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
12 ovex 7464 . . . . . . . 8 (𝑦𝐹𝑧) ∈ V
13 imaeqexov.1 . . . . . . . 8 (𝑥 = (𝑦𝐹𝑧) → (𝜑𝜓))
1412, 13ceqsexv 3530 . . . . . . 7 (∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ 𝜓)
1514rexbii 3092 . . . . . 6 (∃𝑧𝐶𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑧𝐶 𝜓)
1611, 15bitr3i 277 . . . . 5 (∃𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑧𝐶 𝜓)
1716rexbii 3092 . . . 4 (∃𝑦𝐵𝑥𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦𝐵𝑧𝐶 𝜓)
1810, 17bitr3i 277 . . 3 (∃𝑥𝑦𝐵𝑧𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦𝐵𝑧𝐶 𝜓)
199, 18bitrdi 287 . 2 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑦𝐵𝑧𝐶 𝜓))
201, 19bitrid 283 1 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦𝐵𝑧𝐶 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  wss 3963   × cxp 5687  cima 5692   Fn wfn 6558  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434
This theorem is referenced by:  naddunif  8730
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