Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnvexg | GIF version |
Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Ref | Expression |
---|---|
cnvexg | ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4912 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 5054 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4545 | . . . 4 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | rnexg 4799 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
6 | 4, 5 | eqeltrrid 2225 | . . 3 ⊢ (𝐴 ∈ 𝑉 → dom ◡𝐴 ∈ V) |
7 | dfdm4 4726 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
8 | dmexg 4798 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) | |
9 | 7, 8 | eqeltrrid 2225 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran ◡𝐴 ∈ V) |
10 | xpexg 4648 | . . 3 ⊢ ((dom ◡𝐴 ∈ V ∧ ran ◡𝐴 ∈ V) → (dom ◡𝐴 × ran ◡𝐴) ∈ V) | |
11 | 6, 9, 10 | syl2anc 408 | . 2 ⊢ (𝐴 ∈ 𝑉 → (dom ◡𝐴 × ran ◡𝐴) ∈ V) |
12 | ssexg 4062 | . 2 ⊢ ((◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) ∧ (dom ◡𝐴 × ran ◡𝐴) ∈ V) → ◡𝐴 ∈ V) | |
13 | 3, 11, 12 | sylancr 410 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 × cxp 4532 ◡ccnv 4533 dom cdm 4534 ran crn 4535 Rel wrel 4539 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: cnvex 5072 relcnvexb 5073 cofunex2g 6003 cnvf1o 6115 brtpos2 6141 tposexg 6148 cnven 6695 cnvct 6696 fopwdom 6723 relcnvfi 6822 ennnfonelemim 11926 |
Copyright terms: Public domain | W3C validator |