Theorem ralrnmptw 7068

Description: A restricted quantifier over an image set. Version of ralrnmpt 7070 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2371. (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 6660	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
3		dfsbcq 3757	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓))
4	3	ralrn 7062	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
5	2, 4	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
6		nfsbc1v 3775	. . . 4 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
7		nfv 1914	. . . 4 ⊢ Ⅎ𝑤𝜓
8		sbceq2a 3767	. . . 4 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓 ↔ 𝜓))
9	6, 7, 8	cbvralw 3282	. . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓)
10		nfmpt1 5208	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
11	1, 10	nfcxfr 2890	. . . . . 6 ⊢ Ⅎ𝑥𝐹
12		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑥𝑧
13	11, 12	nffv 6870	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
14		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥𝜓
15	13, 14	nfsbcw 3777	. . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓
16		nfv 1914	. . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓
17		fveq2 6860	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18	17	sbceq1d 3760	. . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓))
19	15, 16, 18	cbvralw 3282	. . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)
20	5, 9, 19	3bitr3g 313	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓))
21	1	fvmpt2 6981	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
22	21	sbceq1d 3760	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓))
23		ralrnmptw.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
24	23	sbcieg 3795	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
25	24	adantl 481	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	22, 25	bitrd 279	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
27	26	ralimiaa 3066	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
28		ralbi 3086	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
29	27, 28	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
30	20, 29	bitrd 279	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  [wsbc 3755   ↦ cmpt 5190  ran crn 5641   Fn wfn 6508  ‘cfv 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-fv 6521

This theorem is referenced by:  rexrnmptw  7069  ac6num  10438  gsumwspan  18779  dfod2  19500  ordtbaslem  23081  ordtrest2lem  23096  cncmp  23285  comppfsc  23425  ptpjopn  23505  ordthmeolem  23694  tsmsfbas  24021  tsmsf1o  24038  prdsxmetlem  24262  prdsbl  24385  metdsf  24743  metdsge  24744  minveclem1  25330  minveclem3b  25334  minveclem6  25340  mbflimsup  25573  xrlimcnp  26884  minvecolem1  30809  minvecolem5  30816  minvecolem6  30817  ordtrest2NEWlem  33918  cvmsss2  35261  fin2so  37596  prdsbnd  37782  rrnequiv  37824  ralrnmpt3  45246