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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 37254. (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 5464 | . . . 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 3751 | . . . 4 ⊢ (𝜑 → ((𝑅 “ {𝐴}) ∪ (𝑅 “ 𝑈)) ⊆ 𝑈) |
6 | 2, 5 | syl5eqss 3612 | . . 3 ⊢ (𝜑 → (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) |
7 | trclimalb2 37037 | . . 3 ⊢ ((𝑅 ∈ V ∧ (𝑅 “ ({𝐴} ∪ 𝑈)) ⊆ 𝑈) → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) | |
8 | 1, 6, 7 | syl2anc 691 | . 2 ⊢ (𝜑 → ((t+‘𝑅) “ {𝐴}) ⊆ 𝑈) |
9 | frege77d.ab | . . . 4 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | |
10 | df-br 4584 | . . . 4 ⊢ (𝐴(t+‘𝑅)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) | |
11 | 9, 10 | sylib 207 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (t+‘𝑅)) |
12 | frege77d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) | |
13 | frege77d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
14 | elimasng 5410 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) | |
15 | 12, 13, 14 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((t+‘𝑅) “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ (t+‘𝑅))) |
16 | 11, 15 | mpbird 246 | . 2 ⊢ (𝜑 → 𝐵 ∈ ((t+‘𝑅) “ {𝐴})) |
17 | 8, 16 | sseldd 3569 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 {csn 4125 〈cop 4131 class class class wbr 4583 “ cima 5041 ‘cfv 5804 t+ctcl 13572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-trcl 13574 df-relexp 13609 |
This theorem is referenced by: frege81d 37058 frege87d 37061 |
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