Theorem nfrelp 45610

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 45604	. 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 6705	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴⟶𝐵
6		nfcv 2932	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfrelp.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2932	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5163	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6895	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfrelp.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6895	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5163	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfim 1924	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3319	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3319	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1927	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1881	1 ⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1811  Ⅎwnfc 2917  ∀wral 3086   class class class wbr 5114  ⟶wf 6536  ‘cfv 6540   RelPres wrelp 45603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-relp 45604

This theorem is referenced by: (None)