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Mirrors > Home > NFE Home > Th. List > fununiq | GIF version |
Description: Implicational form of part of the definition of a function. (Contributed by SF, 24-Feb-2015.) |
Ref | Expression |
---|---|
fununiq | ⊢ ((Fun F ∧ AFB ∧ AFC) → B = C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . . . 5 ⊢ (AFB → (A ∈ V ∧ B ∈ V)) | |
2 | brex 4689 | . . . . 5 ⊢ (AFC → (A ∈ V ∧ C ∈ V)) | |
3 | 1, 2 | anim12i 549 | . . . 4 ⊢ ((AFB ∧ AFC) → ((A ∈ V ∧ B ∈ V) ∧ (A ∈ V ∧ C ∈ V))) |
4 | anandi 801 | . . . 4 ⊢ ((A ∈ V ∧ (B ∈ V ∧ C ∈ V)) ↔ ((A ∈ V ∧ B ∈ V) ∧ (A ∈ V ∧ C ∈ V))) | |
5 | 3, 4 | sylibr 203 | . . 3 ⊢ ((AFB ∧ AFC) → (A ∈ V ∧ (B ∈ V ∧ C ∈ V))) |
6 | 5 | 3adant1 973 | . 2 ⊢ ((Fun F ∧ AFB ∧ AFC) → (A ∈ V ∧ (B ∈ V ∧ C ∈ V))) |
7 | dffun2 5119 | . . . . . 6 ⊢ (Fun F ↔ ∀x∀y∀z((xFy ∧ xFz) → y = z)) | |
8 | breq12 4644 | . . . . . . . . . 10 ⊢ ((x = A ∧ y = B) → (xFy ↔ AFB)) | |
9 | 8 | 3adant3 975 | . . . . . . . . 9 ⊢ ((x = A ∧ y = B ∧ z = C) → (xFy ↔ AFB)) |
10 | breq12 4644 | . . . . . . . . . 10 ⊢ ((x = A ∧ z = C) → (xFz ↔ AFC)) | |
11 | 10 | 3adant2 974 | . . . . . . . . 9 ⊢ ((x = A ∧ y = B ∧ z = C) → (xFz ↔ AFC)) |
12 | 9, 11 | anbi12d 691 | . . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = C) → ((xFy ∧ xFz) ↔ (AFB ∧ AFC))) |
13 | eqeq12 2365 | . . . . . . . . 9 ⊢ ((y = B ∧ z = C) → (y = z ↔ B = C)) | |
14 | 13 | 3adant1 973 | . . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = C) → (y = z ↔ B = C)) |
15 | 12, 14 | imbi12d 311 | . . . . . . 7 ⊢ ((x = A ∧ y = B ∧ z = C) → (((xFy ∧ xFz) → y = z) ↔ ((AFB ∧ AFC) → B = C))) |
16 | 15 | spc3gv 2944 | . . . . . 6 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (∀x∀y∀z((xFy ∧ xFz) → y = z) → ((AFB ∧ AFC) → B = C))) |
17 | 7, 16 | syl5bi 208 | . . . . 5 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (Fun F → ((AFB ∧ AFC) → B = C))) |
18 | 17 | exp4a 589 | . . . 4 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (Fun F → (AFB → (AFC → B = C)))) |
19 | 18 | 3impd 1165 | . . 3 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → ((Fun F ∧ AFB ∧ AFC) → B = C)) |
20 | 19 | 3expb 1152 | . 2 ⊢ ((A ∈ V ∧ (B ∈ V ∧ C ∈ V)) → ((Fun F ∧ AFB ∧ AFC) → B = C)) |
21 | 6, 20 | mpcom 32 | 1 ⊢ ((Fun F ∧ AFB ∧ AFC) → B = C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∀wal 1540 = wceq 1642 ∈ wcel 1710 Vcvv 2859 class class class wbr 4639 Fun wfun 4775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-id 4767 df-cnv 4785 df-fun 4789 |
This theorem is referenced by: funsi 5520 fntxp 5804 fnpprod 5843 enpw1 6062 enprmaplem3 6078 |
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