Theorem nfrelp 46032

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 46026	. 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 2922	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfrelp.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5151	. . . . . 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 5151	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfim 1928	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3309	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3309	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1931	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1885	1 ⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1815  Ⅎwnfc 2907  ∀wral 3076   class class class wbr 5102  ⟶wf 6534  ‘cfv 6538   RelPres wrelp 46025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-relp 46026

This theorem is used by: (None)