Theorem oppfvallem 49121

Description: Lemma for oppfval 49122. (Contributed by Zhi Wang, 13-Nov-2025.)

Assertion

Ref	Expression
oppfvallem	⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (Rel 𝐺 ∧ Rel dom 𝐺))

Proof of Theorem oppfvallem

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		id 22	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
3	1, 2	funcfn2 17794	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
4		fnrel 6588	. . 3 ⊢ (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) → Rel 𝐺)
5	3, 4	syl 17	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → Rel 𝐺)
6		relxp 5641	. . 3 ⊢ Rel ((Base‘𝐶) × (Base‘𝐶))
7	3	fndmd 6591	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → dom 𝐺 = ((Base‘𝐶) × (Base‘𝐶)))
8	7	releqd 5726	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (Rel dom 𝐺 ↔ Rel ((Base‘𝐶) × (Base‘𝐶))))
9	6, 8	mpbiri 258	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → Rel dom 𝐺)
10	5, 9	jca 511	1 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (Rel 𝐺 ∧ Rel dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   class class class wbr 5095   × cxp 5621  dom cdm 5623  Rel wrel 5628   Fn wfn 6481  ‘cfv 6486  (class class class)co 7353  Basecbs 17138   Func cfunc 17779

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  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  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-ov 7356  df-oprab 7357  df-mpo 7358  df-map 8762  df-ixp 8832  df-func 17783

This theorem is referenced by:  oppfval  49122