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| Mirrors > Home > MPE Home > Th. List > fndmeng | Structured version Visualization version GIF version | ||
| Description: A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
| Ref | Expression |
|---|---|
| fndmeng | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnex 7237 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V) | |
| 2 | fnfun 6668 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → Fun 𝐹) |
| 4 | fundmeng 9072 | . . 3 ⊢ ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹) | |
| 5 | 1, 3, 4 | syl2anc 584 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → dom 𝐹 ≈ 𝐹) |
| 6 | fndm 6671 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 7 | 6 | breq1d 5153 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹)) |
| 8 | 7 | adantr 480 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹)) |
| 9 | 5, 8 | mpbid 232 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-en 8986 |
| This theorem is referenced by: tskcard 10821 hashfn 14414 |
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