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Mirrors > Home > MPE Home > Th. List > fvimacnv | Structured version Visualization version GIF version |
Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6430 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
fvimacnv | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 6812 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | fvex 6676 | . . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V | |
3 | opelcnvg 5744 | . . . . . . 7 ⊢ (((𝐹‘𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
4 | 2, 3 | mpan 686 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → (〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) |
6 | 1, 5 | mpbird 258 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹) |
7 | elimasng 5948 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹)) | |
8 | 2, 7 | mpan 686 | . . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹)) |
9 | 8 | adantl 482 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹)) |
10 | 6, 9 | mpbird 258 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)})) |
11 | 2 | snss 4710 | . . . . 5 ⊢ ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵) |
12 | imass2 5958 | . . . . 5 ⊢ ({(𝐹‘𝐴)} ⊆ 𝐵 → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵)) | |
13 | 11, 12 | sylbi 218 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵)) |
14 | 13 | sseld 3963 | . . 3 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
15 | 10, 14 | syl5com 31 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
16 | fvimacnvi 6814 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) | |
17 | 16 | ex 413 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)) |
18 | 17 | adantr 481 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)) |
19 | 15, 18 | impbid 213 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {csn 4557 〈cop 4563 ◡ccnv 5547 dom cdm 5548 “ cima 5551 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 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 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-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: funimass3 6816 elpreima 6820 iinpreima 6829 isr0 22273 rnelfmlem 22488 rnelfm 22489 fmfnfmlem2 22491 fmfnfmlem4 22493 fmfnfm 22494 metustid 23091 metustsym 23092 metustexhalf 23093 xppreima 30322 dstfrvel 31630 ballotlemrv 31676 bj-fvimacnv0 34456 grpokerinj 35052 diaintclN 38074 dibintclN 38183 dihintcl 38360 arearect 39700 areaquad 39701 |
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