Theorem ralrnmptw 7115

Description: A restricted quantifier over an image set. Version of ralrnmpt 7117 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝑦   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 ralrnmptw

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

Step	Hyp	Ref	Expression
1		ralrnmptw.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 6698	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
3		dfsbcq 3744	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓))
4	3	ralrn 7109	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
5	2, 4	syl 18	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
6		nfsbc1v 3762	. . . 4 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
7		nfv 1949	. . . 4 ⊢ Ⅎ𝑤𝜓
8		sbceq2a 3754	. . . 4 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓 ↔ 𝜓))
9	6, 7, 8	cbvralw 3308	. . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
10		nfmpt1 5207	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
11	1, 10	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥𝑧
13	11, 12	nffv 6915	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
14		nfv 1949	. . . . 5 ⊢ Ⅎ𝑥𝜓
15	13, 14	nfsbcw 3764	. . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓
16		nfv 1949	. . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓
17		fveq2 6905	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18	17	sbceq1d 3747	. . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓))
19	15, 16, 18	cbvralw 3308	. . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)
20	5, 9, 19	3bitr3g 316	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓))
21	1	fvmpt2 7025	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
22	21	sbceq1d 3747	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓))
23		ralrnmptw.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
24	23	sbcieg 3781	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
25	24	adantl 487	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	22, 25	bitrd 282	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
27	26	ralimiaa 3102	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
28		ralbi 3121	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
29	27, 28	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
30	20, 29	bitrd 282	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  [wsbc 3742   ↦ cmpt 5189  ran crn 5656   Fn wfn 6550  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-fv 6563

This theorem is used by:  rexrnmptw  7116  ac6num  10581  gsumwspan  19068  dfod2  19805  ordtbaslem  23533  ordtrest2lem  23548  cncmp  23737  comppfsc  23878  ptpjopn  23958  ordthmeolem  24147  tsmsfbas  24474  tsmsf1o  24491  prdsxmetlem  24714  prdsbl  24837  metdsf  25195  metdsge  25196  minveclem1  25772  minveclem3b  25776  minveclem6  25782  mbflimsup  26014  xrlimcnp  27323  minvecolem1  31503  minvecolem5  31510  minvecolem6  31511  ordtrest2NEWlem  34621  cvmsss2  36205  fin2so  38721  prdsbnd  38918  rrnequiv  38960  ralrnmpt3  46474