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Mirrors > Home > ILE Home > Th. List > fundmeng | GIF version |
Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.) |
Ref | Expression |
---|---|
fundmeng | ⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeq 5202 | . . . 4 ⊢ (𝑥 = 𝐹 → (Fun 𝑥 ↔ Fun 𝐹)) | |
2 | dmeq 4798 | . . . . 5 ⊢ (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹) | |
3 | id 19 | . . . . 5 ⊢ (𝑥 = 𝐹 → 𝑥 = 𝐹) | |
4 | 2, 3 | breq12d 3989 | . . . 4 ⊢ (𝑥 = 𝐹 → (dom 𝑥 ≈ 𝑥 ↔ dom 𝐹 ≈ 𝐹)) |
5 | 1, 4 | imbi12d 233 | . . 3 ⊢ (𝑥 = 𝐹 → ((Fun 𝑥 → dom 𝑥 ≈ 𝑥) ↔ (Fun 𝐹 → dom 𝐹 ≈ 𝐹))) |
6 | vex 2724 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6 | fundmen 6763 | . . 3 ⊢ (Fun 𝑥 → dom 𝑥 ≈ 𝑥) |
8 | 5, 7 | vtoclg 2781 | . 2 ⊢ (𝐹 ∈ 𝑉 → (Fun 𝐹 → dom 𝐹 ≈ 𝐹)) |
9 | 8 | imp 123 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 dom cdm 4598 Fun wfun 5176 ≈ cen 6695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-en 6698 |
This theorem is referenced by: fndmeng 6767 fundmfi 6894 |
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