Theorem dffrege76 38233

Description: If from the two propositions that every result of an application of the procedure 𝑅 to 𝐵 has property 𝑓 and that property 𝑓 is hereditary in the 𝑅-sequence, it can be inferred, whatever 𝑓 may be, that 𝐸 has property 𝑓, then we say 𝐸 follows 𝐵 in the 𝑅-sequence. Definition 76 of [Frege1879] p. 60.

Each of 𝐵, 𝐸 and 𝑅 must be sets. (Contributed by RP, 2-Jul-2020.)

Hypotheses

Ref	Expression
frege76.b	⊢ 𝐵 ∈ 𝑈
frege76.e	⊢ 𝐸 ∈ 𝑉
frege76.r	⊢ 𝑅 ∈ 𝑊

Assertion

Ref	Expression
dffrege76	⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)

Distinct variable groups:   𝑓,𝑎,𝐵   𝑓,𝐸   𝑅,𝑎,𝑓   𝑈,𝑓   𝑓,𝑉   𝑓,𝑊

Allowed substitution hints:   𝑈(𝑎)   𝐸(𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem dffrege76

Step	Hyp	Ref	Expression
1		frege76.b	. . 3 ⊢ 𝐵 ∈ 𝑈
2		frege76.e	. . 3 ⊢ 𝐸 ∈ 𝑉
3		frege76.r	. . 3 ⊢ 𝑅 ∈ 𝑊
4		brtrclfv2 38019	. . 3 ⊢ ((𝐵 ∈ 𝑈 ∧ 𝐸 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐵(t+‘𝑅)𝐸 ↔ 𝐸 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓}))
5	1, 2, 3, 4	mp3an 1424	. 2 ⊢ (𝐵(t+‘𝑅)𝐸 ↔ 𝐸 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓})
6	2	elexi 3213	. . 3 ⊢ 𝐸 ∈ V
7	6	elintab 4487	. 2 ⊢ (𝐸 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓))
8		imaundi 5545	. . . . . . . . 9 ⊢ (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅 “ {𝐵}) ∪ (𝑅 “ 𝑓))
9	8	equncomi 3759	. . . . . . . 8 ⊢ (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅 “ 𝑓) ∪ (𝑅 “ {𝐵}))
10	9	sseq1i 3629	. . . . . . 7 ⊢ ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ 𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓)
11		unss 3787	. . . . . . 7 ⊢ (((𝑅 “ 𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ ((𝑅 “ 𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓)
12	10, 11	bitr4i 267	. . . . . 6 ⊢ ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ 𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓))
13		df-he 38067	. . . . . . . 8 ⊢ (𝑅 hereditary 𝑓 ↔ (𝑅 “ 𝑓) ⊆ 𝑓)
14	13	bicomi 214	. . . . . . 7 ⊢ ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ 𝑅 hereditary 𝑓)
15		dfss2 3591	. . . . . . . 8 ⊢ ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎 ∈ 𝑓))
16	1	elexi 3213	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
17		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
18	16, 17	elimasn 5490	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝑎⟩ ∈ 𝑅)
19		df-br 4654	. . . . . . . . . . 11 ⊢ (𝐵𝑅𝑎 ↔ ⟨𝐵, 𝑎⟩ ∈ 𝑅)
20	18, 19	bitr4i 267	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑅 “ {𝐵}) ↔ 𝐵𝑅𝑎)
21	20	imbi1i 339	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎 ∈ 𝑓) ↔ (𝐵𝑅𝑎 → 𝑎 ∈ 𝑓))
22	21	albii 1747	. . . . . . . 8 ⊢ (∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓))
23	15, 22	bitri 264	. . . . . . 7 ⊢ ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓))
24	14, 23	anbi12i 733	. . . . . 6 ⊢ (((𝑅 “ 𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)))
25	12, 24	bitri 264	. . . . 5 ⊢ ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)))
26	25	imbi1i 339	. . . 4 ⊢ (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓) ↔ ((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) → 𝐸 ∈ 𝑓))
27		impexp 462	. . . 4 ⊢ (((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) → 𝐸 ∈ 𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)))
28	26, 27	bitri 264	. . 3 ⊢ (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)))
29	28	albii 1747	. 2 ⊢ (∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓) ↔ ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)))
30	5, 7, 29	3bitrri 287	1 ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   ∈ wcel 1990  {cab 2608   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653   “ cima 5117  ‘cfv 5888  t+ctcl 13724   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-trcl 13726  df-relexp 13761  df-he 38067

This theorem is referenced by:  frege77  38234  frege89  38246