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Mirrors > Home > MPE Home > Th. List > funfvop | Structured version Visualization version GIF version |
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.) |
Ref | Expression |
---|---|
funfvop | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . 2 ⊢ (𝐹‘𝐴) = (𝐹‘𝐴) | |
2 | funopfvb 6714 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = (𝐹‘𝐴) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
3 | 1, 2 | mpbii 235 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 〈cop 4565 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: funfvbrb 6814 fvimacnv 6816 fnopfv 6836 fvelrn 6837 dff3 6859 fnsnb 6921 funfvima3 6990 wfrlem17 7963 tfrlem9a 8014 fundmen 8575 adj1 29702 fgreu 30409 fnsnbt 39111 |
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