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Mirrors > Home > ILE Home > Th. List > funcnveq | GIF version |
Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 4855. (Contributed by Jim Kingdon, 24-Dec-2018.) |
Ref | Expression |
---|---|
funcnveq | ⊢ (Fun ◡A ↔ ∀x∀y∀z((xAy ∧ zAy) → x = z)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4646 | . . 3 ⊢ Rel ◡A | |
2 | dffun2 4855 | . . 3 ⊢ (Fun ◡A ↔ (Rel ◡A ∧ ∀y∀x∀z((y◡Ax ∧ y◡Az) → x = z))) | |
3 | 1, 2 | mpbiran 846 | . 2 ⊢ (Fun ◡A ↔ ∀y∀x∀z((y◡Ax ∧ y◡Az) → x = z)) |
4 | alcom 1364 | . 2 ⊢ (∀y∀x∀z((y◡Ax ∧ y◡Az) → x = z) ↔ ∀x∀y∀z((y◡Ax ∧ y◡Az) → x = z)) | |
5 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
6 | vex 2554 | . . . . . . 7 ⊢ x ∈ V | |
7 | 5, 6 | brcnv 4461 | . . . . . 6 ⊢ (y◡Ax ↔ xAy) |
8 | vex 2554 | . . . . . . 7 ⊢ z ∈ V | |
9 | 5, 8 | brcnv 4461 | . . . . . 6 ⊢ (y◡Az ↔ zAy) |
10 | 7, 9 | anbi12i 433 | . . . . 5 ⊢ ((y◡Ax ∧ y◡Az) ↔ (xAy ∧ zAy)) |
11 | 10 | imbi1i 227 | . . . 4 ⊢ (((y◡Ax ∧ y◡Az) → x = z) ↔ ((xAy ∧ zAy) → x = z)) |
12 | 11 | 2albii 1357 | . . 3 ⊢ (∀y∀z((y◡Ax ∧ y◡Az) → x = z) ↔ ∀y∀z((xAy ∧ zAy) → x = z)) |
13 | 12 | albii 1356 | . 2 ⊢ (∀x∀y∀z((y◡Ax ∧ y◡Az) → x = z) ↔ ∀x∀y∀z((xAy ∧ zAy) → x = z)) |
14 | 3, 4, 13 | 3bitri 195 | 1 ⊢ (Fun ◡A ↔ ∀x∀y∀z((xAy ∧ zAy) → x = z)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 class class class wbr 3755 ◡ccnv 4287 Rel wrel 4293 Fun wfun 4839 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-fun 4847 |
This theorem is referenced by: imain 4924 |
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