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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 7168 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V) | |
2 | fnfun 6603 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | 2 | adantr 482 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → Fun 𝐹) |
4 | fundmeng 8977 | . . 3 ⊢ ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹) | |
5 | 1, 3, 4 | syl2anc 585 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → dom 𝐹 ≈ 𝐹) |
6 | fndm 6606 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | breq1d 5116 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹)) |
8 | 7 | adantr 482 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹)) |
9 | 5, 8 | mpbid 231 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3446 class class class wbr 5106 dom cdm 5634 Fun wfun 6491 Fn wfn 6492 ≈ cen 8881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-en 8885 |
This theorem is referenced by: tskcard 10718 hashfn 14276 |
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