Theorem nfrelp 44959

Description: Bound-variable hypothesis builder for a relation-preserving function. (Contributed by Eric Schmidt, 11-Oct-2025.)

Hypotheses

Ref	Expression
nfrelp.1	⊢ Ⅎ𝑥𝐻
nfrelp.2	⊢ Ⅎ𝑥𝑅
nfrelp.3	⊢ Ⅎ𝑥𝑆
nfrelp.4	⊢ Ⅎ𝑥𝐴
nfrelp.5	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfrelp	⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)

Proof of Theorem nfrelp

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

Step	Hyp	Ref	Expression
1		df-relp 44953	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))))
2		nfrelp.1	. . . 4 ⊢ Ⅎ𝑥𝐻
3		nfrelp.4	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfrelp.5	. . . 4 ⊢ Ⅎ𝑥𝐵
5	2, 3, 4	nff 6740	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴⟶𝐵
6		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfrelp.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5198	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6924	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfrelp.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6924	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5198	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfim 1896	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3311	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3311	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1899	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1852	1 ⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1782  Ⅎwnfc 2890  ∀wral 3061   class class class wbr 5151  ⟶wf 6565  ‘cfv 6569   RelPres wrelp 44952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577  df-relp 44953

This theorem is referenced by: (None)