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Mirrors > Home > NFE Home > Th. List > fvclss | GIF version |
Description: Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.) |
Ref | Expression |
---|---|
fvclss | ⊢ {y ∣ ∃x y = (F ‘x)} ⊆ (ran F ∪ {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2355 | . . . . . . . . . 10 ⊢ (y = (F ‘x) ↔ (F ‘x) = y) | |
2 | tz6.12i 5348 | . . . . . . . . . 10 ⊢ (y ≠ ∅ → ((F ‘x) = y → xFy)) | |
3 | 1, 2 | syl5bi 208 | . . . . . . . . 9 ⊢ (y ≠ ∅ → (y = (F ‘x) → xFy)) |
4 | 3 | eximdv 1622 | . . . . . . . 8 ⊢ (y ≠ ∅ → (∃x y = (F ‘x) → ∃x xFy)) |
5 | 4 | com12 27 | . . . . . . 7 ⊢ (∃x y = (F ‘x) → (y ≠ ∅ → ∃x xFy)) |
6 | elrn 4896 | . . . . . . 7 ⊢ (y ∈ ran F ↔ ∃x xFy) | |
7 | 5, 6 | syl6ibr 218 | . . . . . 6 ⊢ (∃x y = (F ‘x) → (y ≠ ∅ → y ∈ ran F)) |
8 | 7 | necon1bd 2584 | . . . . 5 ⊢ (∃x y = (F ‘x) → (¬ y ∈ ran F → y = ∅)) |
9 | vex 2862 | . . . . . 6 ⊢ y ∈ V | |
10 | 9 | elsnc 3756 | . . . . 5 ⊢ (y ∈ {∅} ↔ y = ∅) |
11 | 8, 10 | syl6ibr 218 | . . . 4 ⊢ (∃x y = (F ‘x) → (¬ y ∈ ran F → y ∈ {∅})) |
12 | 11 | orrd 367 | . . 3 ⊢ (∃x y = (F ‘x) → (y ∈ ran F ∨ y ∈ {∅})) |
13 | elun 3220 | . . 3 ⊢ (y ∈ (ran F ∪ {∅}) ↔ (y ∈ ran F ∨ y ∈ {∅})) | |
14 | 12, 13 | sylibr 203 | . 2 ⊢ (∃x y = (F ‘x) → y ∈ (ran F ∪ {∅})) |
15 | 14 | abssi 3341 | 1 ⊢ {y ∣ ∃x y = (F ‘x)} ⊆ (ran F ∪ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 357 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ≠ wne 2516 ∪ cun 3207 ⊆ wss 3257 ∅c0 3550 {csn 3737 class class class wbr 4639 ran crn 4773 ‘cfv 4781 |
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-br 4640 df-ima 4727 df-rn 4786 df-fv 4795 |
This theorem is referenced by: (None) |
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