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Mirrors > Home > MPE Home > Th. List > Mathboxes > brfvidRP | Structured version Visualization version GIF version |
Description: If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 41015. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
brfvidRP.r | ⊢ (𝜑 → 𝑅 ∈ V) |
Ref | Expression |
---|---|
brfvidRP | ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfid6 14623 | . . 3 ⊢ I = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛)) | |
2 | brfvidRP.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
3 | 1nn0 12135 | . . . 4 ⊢ 1 ∈ ℕ0 | |
4 | snssi 4737 | . . . 4 ⊢ (1 ∈ ℕ0 → {1} ⊆ ℕ0) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝜑 → {1} ⊆ ℕ0) |
6 | 1, 2, 5 | brmptiunrelexpd 41015 | . 2 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ ∃𝑛 ∈ {1}𝐴(𝑅↑𝑟𝑛)𝐵)) |
7 | oveq2 7242 | . . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)) | |
8 | 7 | breqd 5080 | . . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟1)𝐵)) |
9 | 8 | rexsng 4606 | . . 3 ⊢ (1 ∈ ℕ0 → (∃𝑛 ∈ {1}𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟1)𝐵)) |
10 | 3, 9 | mp1i 13 | . 2 ⊢ (𝜑 → (∃𝑛 ∈ {1}𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟1)𝐵)) |
11 | 2 | relexp1d 14624 | . . 3 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) |
12 | 11 | breqd 5080 | . 2 ⊢ (𝜑 → (𝐴(𝑅↑𝑟1)𝐵 ↔ 𝐴𝑅𝐵)) |
13 | 6, 10, 12 | 3bitrd 308 | 1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃wrex 3065 Vcvv 3423 ⊆ wss 3883 {csn 4557 class class class wbr 5069 I cid 5470 ‘cfv 6400 (class class class)co 7234 1c1 10759 ℕ0cn0 12119 ↑𝑟crelexp 14614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-nn 11860 df-n0 12120 df-z 12206 df-uz 12468 df-seq 13606 df-relexp 14615 |
This theorem is referenced by: (None) |
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