Theorem imaeqexov 7579

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 3057	. 2 ⊢ (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑))
2		ovelimab 7519	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧)))
3	2	anbi1d 631	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
4		r19.41v 3162	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
5	4	rexbii 3079	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
6		r19.41v 3162	. . . . . 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 1922	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
10		rexcom4 3259	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
11		rexcom4 3259	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
12		ovex 7374	. . . . . . . 8 ⊢ (𝑦𝐹𝑧) ∈ V
13		imaeqexov.1	. . . . . . . 8 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
14	12, 13	ceqsexv 3486	. . . . . . 7 ⊢ (∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ 𝜓)
15	14	rexbii 3079	. . . . . 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 3079	. . . 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 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897   × cxp 5609   “ cima 5614   Fn wfn 6471  (class class class)co 7341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484  df-ov 7344

This theorem is referenced by:  naddunif  8603