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Mirrors > Home > NFE Home > Th. List > dfrnf | GIF version |
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
dfrnf.1 | ⊢ ℲxA |
dfrnf.2 | ⊢ ℲyA |
Ref | Expression |
---|---|
dfrnf | ⊢ ran A = {y ∣ ∃x xAy} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 4902 | . 2 ⊢ ran A = {w ∣ ∃v vAw} | |
2 | nfcv 2489 | . . . . 5 ⊢ Ⅎxv | |
3 | dfrnf.1 | . . . . 5 ⊢ ℲxA | |
4 | nfcv 2489 | . . . . 5 ⊢ Ⅎxw | |
5 | 2, 3, 4 | nfbr 4683 | . . . 4 ⊢ Ⅎx vAw |
6 | nfv 1619 | . . . 4 ⊢ Ⅎv xAw | |
7 | breq1 4642 | . . . 4 ⊢ (v = x → (vAw ↔ xAw)) | |
8 | 5, 6, 7 | cbvex 1985 | . . 3 ⊢ (∃v vAw ↔ ∃x xAw) |
9 | 8 | abbii 2465 | . 2 ⊢ {w ∣ ∃v vAw} = {w ∣ ∃x xAw} |
10 | nfcv 2489 | . . . . 5 ⊢ Ⅎyx | |
11 | dfrnf.2 | . . . . 5 ⊢ ℲyA | |
12 | nfcv 2489 | . . . . 5 ⊢ Ⅎyw | |
13 | 10, 11, 12 | nfbr 4683 | . . . 4 ⊢ Ⅎy xAw |
14 | 13 | nfex 1843 | . . 3 ⊢ Ⅎy∃x xAw |
15 | nfv 1619 | . . 3 ⊢ Ⅎw∃x xAy | |
16 | breq2 4643 | . . . 4 ⊢ (w = y → (xAw ↔ xAy)) | |
17 | 16 | exbidv 1626 | . . 3 ⊢ (w = y → (∃x xAw ↔ ∃x xAy)) |
18 | 14, 15, 17 | cbvab 2471 | . 2 ⊢ {w ∣ ∃x xAw} = {y ∣ ∃x xAy} |
19 | 1, 9, 18 | 3eqtri 2377 | 1 ⊢ ran A = {y ∣ ∃x xAy} |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1541 = wceq 1642 {cab 2339 Ⅎwnfc 2476 class class class wbr 4639 ran crn 4773 |
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 |
This theorem is referenced by: rnopab 4967 |
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