Theorem oppfvallem 49720

Description: Lemma for oppfval 49721. (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 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		id 22	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
3	1, 2	funcfn2 17885	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
4		fnrel 6619	. . 3 ⊢ (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) → Rel 𝐺)
5	3, 4	syl 17	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → Rel 𝐺)
6		relxp 5663	. . 3 ⊢ Rel ((Base‘𝐶) × (Base‘𝐶))
7	3	fndmd 6622	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → dom 𝐺 = ((Base‘𝐶) × (Base‘𝐶)))
8	7	releqd 5749	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (Rel dom 𝐺 ↔ Rel ((Base‘𝐶) × (Base‘𝐶))))
9	6, 8	mpbiri 260	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → Rel dom 𝐺)
10	5, 9	jca 519	1 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (Rel 𝐺 ∧ Rel dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 399   class class class wbr 5099   × cxp 5643  dom cdm 5645  Rel wrel 5650   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392  Basecbs 17228   Func cfunc 17870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-map 8805  df-ixp 8876  df-func 17874

This theorem is referenced by:  oppfval  49721