Theorem ralrnmpo 7289

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
ralrnmpo	⊢ (∀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 ralrnmpo

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 7284	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	raleqi 3413	. . 3 ⊢ (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2825	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 3300	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	ralab 3684	. . 3 ⊢ (∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
7		ralcom4 3235	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
8		r19.23v 3279	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
9	8	albii 1820	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
10	7, 9	bitr2i 278	. . 3 ⊢ (∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
11	3, 6, 10	3bitri 299	. 2 ⊢ (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
12		ralcom4 3235	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑))
13		r19.23v 3279	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
14	13	albii 1820	. . . . . 6 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
15	12, 14	bitri 277	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
16		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑧𝜓
17		ralrnmpo.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
18	16, 17	ceqsalg 3529	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓))
19	18	ralimi 3160	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓))
20		ralbi 3167	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓) → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
21	19, 20	syl 17	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
22	15, 21	syl5bbr 287	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
23	22	ralimi 3160	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
24		ralbi 3167	. . 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))
25	23, 24	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
26	11, 25	syl5bb 285	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  ran crn 5556   ∈ cmpo 7158

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-cnv 5563  df-dm 5565  df-rn 5566  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  rexrnmpo  7290  efgval2  18850  txcnp  22228  txcnmpt  22232  txflf  22614