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Theorem funfv1st2nd 8000

Description: The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.)

**Assertion**
Ref	Expression
funfv1st2nd	⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋))

**Proof of Theorem funfv1st2nd**
Step	Hyp	Ref	Expression
1		funrel 6517	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		1st2nd 7993	. . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 581	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		eleq1 2825	. . . . 5 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
5	4	adantl 481	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
6		funopfv 6891	. . . . 5 ⊢ (Fun 𝐹 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
7	6	adantr 480	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
8	5, 7	sylbid 240	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
9	8	impancom 451	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
10	3, 9	mpd 15	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 Rel wrel 5637 Fun wfun 6494 ‘cfv 6500 1^st c1st 7941 2^nd c2nd 7942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6456 df-fun 6502 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: gsumhashmul 33160 satffunlem 35614 satffunlem1lem1 35615 satffunlem2lem1 35617

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