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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege77d | Structured version Visualization version GIF version |
Description: If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 43930. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege77d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege77d.a | ⊢ (𝜑 → 𝐴 ∈ V) |
frege77d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege77d.ab | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
frege77d.he | ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) |
frege77d.ss | ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) |
Ref | Expression |
---|---|
frege77d | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege77d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | imaundi 6172 | . . . 4 ⊢ (𝑅 “ ({𝐴} ∪ 𝑈)) = ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) | |
3 | frege77d.ss | . . . . 5 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) | |
4 | frege77d.he | . . . . 5 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) | |
5 | 3, 4 | unssd 4202 | . . . 4 ⊢ (𝜑 → ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) ⊆ 𝑈) |
6 | 2, 5 | eqsstrid 4044 | . . 3 ⊢ (𝜑 → (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) |
7 | trclimalb2 43716 | . . 3 ⊢ ((𝑅 ∈ V ∧ (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) | |
8 | 1, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) |
9 | frege77d.ab | . . . 4 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | |
10 | df-br 5149 | . . . 4 ⊢ (𝐴(t+‘𝑅)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) | |
11 | 9, 10 | sylib 218 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) |
12 | frege77d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
13 | frege77d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
14 | elimasng 6109 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) | |
15 | 12, 13, 14 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) |
16 | 11, 15 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((t+‘𝑅) “ {𝐴})) |
17 | 8, 16 | sseldd 3996 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 {csn 4631 〈cop 4637 class class class wbr 5148 “ cima 5692 ‘cfv 6563 t+ctcl 15021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-trcl 15023 df-relexp 15056 |
This theorem is referenced by: frege81d 43737 frege87d 43740 |
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