Theorem nfrelp 44923

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 44917	. 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 6652	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴⟶𝐵
6		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfrelp.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5142	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6836	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfrelp.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6836	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5142	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfim 1896	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3277	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3277	. . 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 1853	1 ⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1783  Ⅎwnfc 2876  ∀wral 3044   class class class wbr 5095  ⟶wf 6482  ‘cfv 6486   RelPres wrelp 44916

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 2701

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-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-relp 44917

This theorem is referenced by: (None)