Theorem imaeqexov 7606

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) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
imaeqexov	⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 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

Step	Hyp	Ref	Expression
1		df-rex 3063	. 2 ⊢ (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑))
2		ovelimab 7546	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧)))
3	2	anbi1d 632	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
4		r19.41v 3168	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
5	4	rexbii 3085	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
6		r19.41v 3168	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
7	5, 6	bitr2i 276	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
8	3, 7	bitrdi 287	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
9	8	exbidv 1923	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
10		rexcom4 3265	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
11		rexcom4 3265	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
12		ovex 7401	. . . . . . . 8 ⊢ (𝑦𝐹𝑧) ∈ V
13		imaeqexov.1	. . . . . . . 8 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
14	12, 13	ceqsexv 3492	. . . . . . 7 ⊢ (∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ 𝜓)
15	14	rexbii 3085	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑧 ∈ 𝐶 𝜓)
16	11, 15	bitr3i 277	. . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑧 ∈ 𝐶 𝜓)
17	16	rexbii 3085	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓)
18	10, 17	bitr3i 277	. . 3 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓)
19	9, 18	bitrdi 287	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓))
20	1, 19	bitrid 283	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903   × cxp 5630   “ cima 5635   Fn wfn 6495  (class class class)co 7368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-ov 7371

This theorem is referenced by:  naddunif  8631