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Theorem isf32lem1 10350
Description: Lemma for isfin3-2 10364. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (πœ‘ β†’ 𝐹:Ο‰βŸΆπ’« 𝐺)
isf32lem.b (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯))
isf32lem.c (πœ‘ β†’ Β¬ ∩ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem1 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 βŠ† 𝐴 ∧ πœ‘)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐴   π‘₯,𝐹
Allowed substitution hint:   𝐺(π‘₯)

Proof of Theorem isf32lem1
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . 5 (π‘Ž = 𝐡 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π΅))
21sseq1d 4013 . . . 4 (π‘Ž = 𝐡 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅)))
32imbi2d 340 . . 3 (π‘Ž = 𝐡 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅))))
4 fveq2 6891 . . . . 5 (π‘Ž = 𝑏 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))
54sseq1d 4013 . . . 4 (π‘Ž = 𝑏 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜π‘) βŠ† (πΉβ€˜π΅)))
65imbi2d 340 . . 3 (π‘Ž = 𝑏 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜π‘) βŠ† (πΉβ€˜π΅))))
7 fveq2 6891 . . . . 5 (π‘Ž = suc 𝑏 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜suc 𝑏))
87sseq1d 4013 . . . 4 (π‘Ž = suc 𝑏 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅)))
98imbi2d 340 . . 3 (π‘Ž = suc 𝑏 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅))))
10 fveq2 6891 . . . . 5 (π‘Ž = 𝐴 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π΄))
1110sseq1d 4013 . . . 4 (π‘Ž = 𝐴 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅)))
1211imbi2d 340 . . 3 (π‘Ž = 𝐴 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))))
13 ssid 4004 . . . 4 (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅)
14132a1i 12 . . 3 (𝐡 ∈ Ο‰ β†’ (πœ‘ β†’ (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅)))
15 isf32lem.b . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯))
16 suceq 6430 . . . . . . . . . 10 (π‘₯ = 𝑏 β†’ suc π‘₯ = suc 𝑏)
1716fveq2d 6895 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ (πΉβ€˜suc π‘₯) = (πΉβ€˜suc 𝑏))
18 fveq2 6891 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘))
1917, 18sseq12d 4015 . . . . . . . 8 (π‘₯ = 𝑏 β†’ ((πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯) ↔ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
2019rspcv 3608 . . . . . . 7 (𝑏 ∈ Ο‰ β†’ (βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯) β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
2115, 20syl5 34 . . . . . 6 (𝑏 ∈ Ο‰ β†’ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
2221ad2antrr 724 . . . . 5 (((𝑏 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝑏) β†’ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
23 sstr2 3989 . . . . 5 ((πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘) β†’ ((πΉβ€˜π‘) βŠ† (πΉβ€˜π΅) β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅)))
2422, 23syl6 35 . . . 4 (((𝑏 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝑏) β†’ (πœ‘ β†’ ((πΉβ€˜π‘) βŠ† (πΉβ€˜π΅) β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅))))
2524a2d 29 . . 3 (((𝑏 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝑏) β†’ ((πœ‘ β†’ (πΉβ€˜π‘) βŠ† (πΉβ€˜π΅)) β†’ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅))))
263, 6, 9, 12, 14, 25findsg 7892 . 2 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝐴) β†’ (πœ‘ β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅)))
2726impr 455 1 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 βŠ† 𝐴 ∧ πœ‘)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  π’« cpw 4602  βˆ© cint 4950  ran crn 5677  suc csuc 6366  βŸΆwf 6539  β€˜cfv 6543  Ο‰com 7857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fv 6551  df-om 7858
This theorem is referenced by:  isf32lem2  10351  isf32lem3  10352
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