Theorem rexrnmpt 5310

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
ralrnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
ralrnmpt.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexrnmpt	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝜒,𝑦   𝑦,𝐹   𝜓,𝑥

Allowed substitution hints:   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑦)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem rexrnmpt

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralrnmpt.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 5025	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
3		dfsbcq 2766	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓))
4	3	rexrn 5304	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
5	2, 4	syl 14	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
6		nfv 1421	. . . . 5 ⊢ Ⅎ𝑤𝜓
7		nfsbc1v 2782	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
8		sbceq1a 2773	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
9	6, 7, 8	cbvrex 2530	. . . 4 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
10	9	bicomi 123	. . 3 ⊢ (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑦 ∈ ran 𝐹𝜓)
11		nfmpt1 3850	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12	1, 11	nfcxfr 2175	. . . . . 6 ⊢ Ⅎ𝑥𝐹
13		nfcv 2178	. . . . . 6 ⊢ Ⅎ𝑥𝑧
14	12, 13	nffv 5185	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
15		nfv 1421	. . . . 5 ⊢ Ⅎ𝑥𝜓
16	14, 15	nfsbc 2784	. . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓
17		nfv 1421	. . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓
18		fveq2 5178	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
19	18	sbceq1d 2769	. . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓))
20	16, 17, 19	cbvrex 2530	. . 3 ⊢ (∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)
21	5, 10, 20	3bitr3g 211	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓))
22	1	fvmpt2 5254	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
23	22	sbceq1d 2769	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓))
24		ralrnmpt.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
25	24	sbcieg 2795	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	25	adantl 262	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
27	23, 26	bitrd 177	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
28	27	ralimiaa 2383	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
29		pm5.32 426	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
30	29	albii 1359	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
31		exbi 1495	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
32	30, 31	sylbi 114	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
33		df-ral 2311	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)))
34		df-rex 2312	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓))
35		df-rex 2312	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
36	34, 35	bibi12i 218	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
37	32, 33, 36	3imtr4i 190	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
38	28, 37	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
39	21, 38	bitrd 177	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  [wsbc 2764   ↦ cmpt 3818  ran crn 4346   Fn wfn 4897  ‘cfv 4902

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-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)