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Theorem funfv1st2nd 7992

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 6509	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		1st2nd 7985	. . 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 6883	. . . . 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 4574 Rel wrel 5629 Fun wfun 6486 ‘cfv 6492 1^st c1st 7933 2^nd c2nd 7934

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 5231 ax-nul 5241 ax-pr 5370 ax-un 7682

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 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-1st 7935 df-2nd 7936

This theorem is referenced by: gsumhashmul 33143 satffunlem 35599 satffunlem1lem1 35600 satffunlem2lem1 35602

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