Theorem imaeqexov 7591

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 3054	. 2 ⊢ (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑))
2		ovelimab 7531	. . . . . 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 3159	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
5	4	rexbii 3076	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
6		r19.41v 3159	. . . . . 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 1921	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑)))
10		rexcom4 3256	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
11		rexcom4 3256	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) ∧ 𝜑))
12		ovex 7386	. . . . . . . 8 ⊢ (𝑦𝐹𝑧) ∈ V
13		imaeqexov.1	. . . . . . . 8 ⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))
14	12, 13	ceqsexv 3489	. . . . . . 7 ⊢ (∃𝑥(𝑥 = (𝑦𝐹𝑧) ∧ 𝜑) ↔ 𝜓)
15	14	rexbii 3076	. . . . . 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 3076	. . . 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 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3905   × cxp 5621   “ cima 5626   Fn wfn 6481  (class class class)co 7353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-ov 7356

This theorem is referenced by:  naddunif  8618