New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dminss | GIF version |
Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by set.mm contributors, 11-Aug-2004.) |
Ref | Expression |
---|---|
dminss | ⊢ (dom R ∩ A) ⊆ (◡R “ (R “ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspe 2675 | . . . . . . 7 ⊢ ((x ∈ A ∧ xRy) → ∃x ∈ A xRy) | |
2 | elima 4754 | . . . . . . 7 ⊢ (y ∈ (R “ A) ↔ ∃x ∈ A xRy) | |
3 | 1, 2 | sylibr 203 | . . . . . 6 ⊢ ((x ∈ A ∧ xRy) → y ∈ (R “ A)) |
4 | 3 | ancoms 439 | . . . . 5 ⊢ ((xRy ∧ x ∈ A) → y ∈ (R “ A)) |
5 | brcnv 4892 | . . . . . . 7 ⊢ (y◡Rx ↔ xRy) | |
6 | 5 | biimpri 197 | . . . . . 6 ⊢ (xRy → y◡Rx) |
7 | 6 | adantr 451 | . . . . 5 ⊢ ((xRy ∧ x ∈ A) → y◡Rx) |
8 | 4, 7 | jca 518 | . . . 4 ⊢ ((xRy ∧ x ∈ A) → (y ∈ (R “ A) ∧ y◡Rx)) |
9 | 8 | eximi 1576 | . . 3 ⊢ (∃y(xRy ∧ x ∈ A) → ∃y(y ∈ (R “ A) ∧ y◡Rx)) |
10 | eldm 4898 | . . . . 5 ⊢ (x ∈ dom R ↔ ∃y xRy) | |
11 | 10 | anbi1i 676 | . . . 4 ⊢ ((x ∈ dom R ∧ x ∈ A) ↔ (∃y xRy ∧ x ∈ A)) |
12 | elin 3219 | . . . 4 ⊢ (x ∈ (dom R ∩ A) ↔ (x ∈ dom R ∧ x ∈ A)) | |
13 | 19.41v 1901 | . . . 4 ⊢ (∃y(xRy ∧ x ∈ A) ↔ (∃y xRy ∧ x ∈ A)) | |
14 | 11, 12, 13 | 3bitr4i 268 | . . 3 ⊢ (x ∈ (dom R ∩ A) ↔ ∃y(xRy ∧ x ∈ A)) |
15 | elima2 4755 | . . 3 ⊢ (x ∈ (◡R “ (R “ A)) ↔ ∃y(y ∈ (R “ A) ∧ y◡Rx)) | |
16 | 9, 14, 15 | 3imtr4i 257 | . 2 ⊢ (x ∈ (dom R ∩ A) → x ∈ (◡R “ (R “ A))) |
17 | 16 | ssriv 3277 | 1 ⊢ (dom R ∩ A) ⊆ (◡R “ (R “ A)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ∃wrex 2615 ∩ cin 3208 ⊆ wss 3257 class class class wbr 4639 “ cima 4722 ◡ccnv 4771 dom cdm 4772 |
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-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |