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Mirrors > Home > NFE Home > Th. List > funssres | GIF version |
Description: The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
Ref | Expression |
---|---|
funssres | ⊢ ((Fun F ∧ G ⊆ F) → (F ↾ dom G) = G) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3267 | . . . . . 6 ⊢ (G ⊆ F → (〈x, y〉 ∈ G → 〈x, y〉 ∈ F)) | |
2 | 1 | adantl 452 | . . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ G → 〈x, y〉 ∈ F)) |
3 | opeldm 4910 | . . . . . 6 ⊢ (〈x, y〉 ∈ G → x ∈ dom G) | |
4 | 3 | a1i 10 | . . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ G → x ∈ dom G)) |
5 | 2, 4 | jcad 519 | . . . 4 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ G → (〈x, y〉 ∈ F ∧ x ∈ dom G))) |
6 | funeu2 5132 | . . . . . . . . . . 11 ⊢ ((Fun F ∧ 〈x, y〉 ∈ F) → ∃!y〈x, y〉 ∈ F) | |
7 | eldm2 4899 | . . . . . . . . . . . . 13 ⊢ (x ∈ dom G ↔ ∃y〈x, y〉 ∈ G) | |
8 | 1 | ancrd 537 | . . . . . . . . . . . . . 14 ⊢ (G ⊆ F → (〈x, y〉 ∈ G → (〈x, y〉 ∈ F ∧ 〈x, y〉 ∈ G))) |
9 | 8 | eximdv 1622 | . . . . . . . . . . . . 13 ⊢ (G ⊆ F → (∃y〈x, y〉 ∈ G → ∃y(〈x, y〉 ∈ F ∧ 〈x, y〉 ∈ G))) |
10 | 7, 9 | syl5bi 208 | . . . . . . . . . . . 12 ⊢ (G ⊆ F → (x ∈ dom G → ∃y(〈x, y〉 ∈ F ∧ 〈x, y〉 ∈ G))) |
11 | 10 | imp 418 | . . . . . . . . . . 11 ⊢ ((G ⊆ F ∧ x ∈ dom G) → ∃y(〈x, y〉 ∈ F ∧ 〈x, y〉 ∈ G)) |
12 | eupick 2267 | . . . . . . . . . . 11 ⊢ ((∃!y〈x, y〉 ∈ F ∧ ∃y(〈x, y〉 ∈ F ∧ 〈x, y〉 ∈ G)) → (〈x, y〉 ∈ F → 〈x, y〉 ∈ G)) | |
13 | 6, 11, 12 | syl2an 463 | . . . . . . . . . 10 ⊢ (((Fun F ∧ 〈x, y〉 ∈ F) ∧ (G ⊆ F ∧ x ∈ dom G)) → (〈x, y〉 ∈ F → 〈x, y〉 ∈ G)) |
14 | 13 | exp43 595 | . . . . . . . . 9 ⊢ (Fun F → (〈x, y〉 ∈ F → (G ⊆ F → (x ∈ dom G → (〈x, y〉 ∈ F → 〈x, y〉 ∈ G))))) |
15 | 14 | com23 72 | . . . . . . . 8 ⊢ (Fun F → (G ⊆ F → (〈x, y〉 ∈ F → (x ∈ dom G → (〈x, y〉 ∈ F → 〈x, y〉 ∈ G))))) |
16 | 15 | imp 418 | . . . . . . 7 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ F → (x ∈ dom G → (〈x, y〉 ∈ F → 〈x, y〉 ∈ G)))) |
17 | 16 | com34 77 | . . . . . 6 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ F → (〈x, y〉 ∈ F → (x ∈ dom G → 〈x, y〉 ∈ G)))) |
18 | 17 | pm2.43d 44 | . . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ F → (x ∈ dom G → 〈x, y〉 ∈ G))) |
19 | 18 | imp3a 420 | . . . 4 ⊢ ((Fun F ∧ G ⊆ F) → ((〈x, y〉 ∈ F ∧ x ∈ dom G) → 〈x, y〉 ∈ G)) |
20 | 5, 19 | impbid 183 | . . 3 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ G ↔ (〈x, y〉 ∈ F ∧ x ∈ dom G))) |
21 | opelres 4950 | . . 3 ⊢ (〈x, y〉 ∈ (F ↾ dom G) ↔ (〈x, y〉 ∈ F ∧ x ∈ dom G)) | |
22 | 20, 21 | syl6rbbr 255 | . 2 ⊢ ((Fun F ∧ G ⊆ F) → (〈x, y〉 ∈ (F ↾ dom G) ↔ 〈x, y〉 ∈ G)) |
23 | 22 | eqrelrdv 4852 | 1 ⊢ ((Fun F ∧ G ⊆ F) → (F ↾ dom G) = G) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃!weu 2204 ⊆ wss 3257 〈cop 4561 dom cdm 4772 ↾ cres 4774 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-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 |
This theorem is referenced by: fun2ssres 5145 funcnvres 5165 funssfv 5343 oprssov 5603 |
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