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Mirrors > Home > MPE Home > Th. List > fvimacnvi | Structured version Visualization version GIF version |
Description: A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.) |
Ref | Expression |
---|---|
fvimacnvi | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4812 | . . 3 ⊢ (𝐴 ∈ (◡𝐹 “ 𝐵) → {𝐴} ⊆ (◡𝐹 “ 𝐵)) | |
2 | funimass2 6650 | . . 3 ⊢ ((Fun 𝐹 ∧ {𝐴} ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ {𝐴}) ⊆ 𝐵) | |
3 | 1, 2 | sylan2 593 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹 “ {𝐴}) ⊆ 𝐵) |
4 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
5 | 4 | snss 4789 | . . 3 ⊢ ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵) |
6 | cnvimass 6101 | . . . . . 6 ⊢ (◡𝐹 “ 𝐵) ⊆ dom 𝐹 | |
7 | 6 | sseli 3990 | . . . . 5 ⊢ (𝐴 ∈ (◡𝐹 “ 𝐵) → 𝐴 ∈ dom 𝐹) |
8 | funfn 6597 | . . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
9 | fnsnfv 6987 | . . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
10 | 8, 9 | sylanb 581 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
11 | 7, 10 | sylan2 593 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
12 | 11 | sseq1d 4026 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → ({(𝐹‘𝐴)} ⊆ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵)) |
13 | 5, 12 | bitrid 283 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → ((𝐹‘𝐴) ∈ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵)) |
14 | 3, 13 | mpbird 257 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 {csn 4630 ◡ccnv 5687 dom cdm 5688 “ cima 5691 Fun wfun 6556 Fn wfn 6557 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: fvimacnv 7072 elpreima 7077 iinpreima 7088 lmhmpreima 21064 mpfind 22148 ofco2 22472 carsggect 34299 bj-fvimacnv0 37268 fcores 47016 |
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