Theorem rexrnmpo 6147

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
ralrnmpo.2	⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexrnmpo	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 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 rexrnmpo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngop.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	rnmpo 6142	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	rexeqi 2736	. . 3 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2238	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 2558	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	rexab 2969	. . 3 ⊢ (∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
7		rexcom4 2827	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
8		r19.41v 2690	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
9	8	exbii 1654	. . . 4 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
10	7, 9	bitr2i 185	. . 3 ⊢ (∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
11	3, 6, 10	3bitri 206	. 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
12		rexcom4 2827	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑))
13		r19.41v 2690	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
14	13	exbii 1654	. . . . . 6 ⊢ (∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
15	12, 14	bitri 184	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
16		ralrnmpo.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
17	16	ceqsexgv 2936	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
18	17	ralimi 2596	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
19		rexbi 2667	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓) → (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
20	18, 19	syl 14	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
21	15, 20	bitr3id 194	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
22	21	ralimi 2596	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
23		rexbi 2667	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓) → (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))
24	22, 23	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))
25	11, 24	bitrid 192	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  ran crn 4732   ∈ cmpo 6030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742  df-oprab 6032  df-mpo 6033

This theorem is referenced by:  eltx  15053  txrest  15070  txlm  15073