New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > fconst2g | GIF version |
Description: A constant function expressed as a cross product. (Contributed by set.mm contributors, 27-Nov-2007.) |
Ref | Expression |
---|---|
fconst2g | ⊢ (B ∈ C → (F:A–→{B} ↔ F = (A × {B}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvconst 5440 | . . . . . . 7 ⊢ ((F:A–→{B} ∧ x ∈ A) → (F ‘x) = B) | |
2 | 1 | adantlr 695 | . . . . . 6 ⊢ (((F:A–→{B} ∧ B ∈ C) ∧ x ∈ A) → (F ‘x) = B) |
3 | fvconst2g 5451 | . . . . . . 7 ⊢ ((B ∈ C ∧ x ∈ A) → ((A × {B}) ‘x) = B) | |
4 | 3 | adantll 694 | . . . . . 6 ⊢ (((F:A–→{B} ∧ B ∈ C) ∧ x ∈ A) → ((A × {B}) ‘x) = B) |
5 | 2, 4 | eqtr4d 2388 | . . . . 5 ⊢ (((F:A–→{B} ∧ B ∈ C) ∧ x ∈ A) → (F ‘x) = ((A × {B}) ‘x)) |
6 | 5 | ralrimiva 2697 | . . . 4 ⊢ ((F:A–→{B} ∧ B ∈ C) → ∀x ∈ A (F ‘x) = ((A × {B}) ‘x)) |
7 | ffn 5223 | . . . . 5 ⊢ (F:A–→{B} → F Fn A) | |
8 | fnconstg 5252 | . . . . 5 ⊢ (B ∈ C → (A × {B}) Fn A) | |
9 | eqfnfv 5392 | . . . . 5 ⊢ ((F Fn A ∧ (A × {B}) Fn A) → (F = (A × {B}) ↔ ∀x ∈ A (F ‘x) = ((A × {B}) ‘x))) | |
10 | 7, 8, 9 | syl2an 463 | . . . 4 ⊢ ((F:A–→{B} ∧ B ∈ C) → (F = (A × {B}) ↔ ∀x ∈ A (F ‘x) = ((A × {B}) ‘x))) |
11 | 6, 10 | mpbird 223 | . . 3 ⊢ ((F:A–→{B} ∧ B ∈ C) → F = (A × {B})) |
12 | 11 | expcom 424 | . 2 ⊢ (B ∈ C → (F:A–→{B} → F = (A × {B}))) |
13 | fconstg 5251 | . . 3 ⊢ (B ∈ C → (A × {B}):A–→{B}) | |
14 | feq1 5210 | . . 3 ⊢ (F = (A × {B}) → (F:A–→{B} ↔ (A × {B}):A–→{B})) | |
15 | 13, 14 | syl5ibrcom 213 | . 2 ⊢ (B ∈ C → (F = (A × {B}) → F:A–→{B})) |
16 | 12, 15 | impbid 183 | 1 ⊢ (B ∈ C → (F:A–→{B} ↔ F = (A × {B}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∀wral 2614 {csn 3737 × cxp 4770 Fn wfn 4776 –→wf 4777 ‘cfv 4781 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-fv 4795 |
This theorem is referenced by: fconst2 5454 fconst5 5455 |
Copyright terms: Public domain | W3C validator |