Theorem nfrelp 45641

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 45635	. 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 6703	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴⟶𝐵
6		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfrelp.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5159	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6893	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfrelp.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6893	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5159	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfim 1926	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3312	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3312	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1929	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1883	1 ⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1813  Ⅎwnfc 2910  ∀wral 3079   class class class wbr 5110  ⟶wf 6534  ‘cfv 6538   RelPres wrelp 45634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-relp 45635

This theorem is referenced by: (None)