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Theorem funssfv 6693

Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

**Assertion**
Ref	Expression
funssfv	⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))

**Proof of Theorem funssfv**
Step	Hyp	Ref	Expression
1		fvres 6691	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹‘𝐴))
2	1	eqcomd 2829	. . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐹‘𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴))
3		funssres 6400	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
4	3	fveq1d 6674	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺‘𝐴))
5	2, 4	sylan9eqr 2880	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
6	5	3impa 1106	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 dom cdm 5557 ↾ cres 5559 Fun wfun 6351 ‘cfv 6357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365

This theorem is referenced by: fviunfun 7648 funelss 7748 funsssuppss 7858 wfrlem12 7968 wfrlem14 7970 tfrlem9 8023 tfrlem11 8026 ac6sfi 8764 axdc3lem2 9875 axdc3lem4 9877 imasvscaval 16813 pserdv 25019 subgruhgredgd 27068 subumgredg2 27069 subupgr 27071 sspn 28515 bnj945 32047 bnj1502 32122 bnj545 32169 bnj548 32171 subfacp1lem2a 32429 subfacp1lem2b 32430 subfacp1lem5 32433 cvmliftlem10 32543 cvmliftlem13 32545 frrlem10 33134