Theorem rexrnmpt2 5616

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)
ralrnmpt2.2	⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexrnmpt2	⊢ (∀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 rexrnmpt2

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	rnmpt2 5611	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	rexeqi 2510	. . 3 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2046	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 2349	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	rexab 2703	. . 3 ⊢ (∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
7		rexcom4 2577	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
8		r19.41v 2466	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
9	8	exbii 1496	. . . 4 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
10	7, 9	bitr2i 174	. . 3 ⊢ (∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
11	3, 6, 10	3bitri 195	. 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
12		rexcom4 2577	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑))
13		r19.41v 2466	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
14	13	exbii 1496	. . . . . 6 ⊢ (∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
15	12, 14	bitri 173	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
16		ralrnmpt2.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
17	16	ceqsexgv 2673	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
18	17	ralimi 2384	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
19		rexbi 2446	. . . . . 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	syl5bbr 183	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
22	21	ralimi 2384	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
23		rexbi 2446	. . 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	syl5bb 181	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  ran crn 4346   ↦ cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-oprab 5516  df-mpt2 5517

This theorem is referenced by: (None)