Theorem frrlem5d 32327

Description: Lemma for founded recursion. The domain of the union of a subset of 𝐵 is a subset of 𝐴. (Contributed by Paul Chapman, 29-Apr-2012.)

Hypotheses

Ref	Expression
frrlem5.1	⊢ 𝑅 Fr 𝐴
frrlem5.2	⊢ 𝑅 Se 𝐴
frrlem5.3	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Assertion

Ref	Expression
frrlem5d	⊢ (𝐶 ⊆ 𝐵 → dom ∪ 𝐶 ⊆ 𝐴)

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐶(𝑥,𝑦,𝑓)

Proof of Theorem frrlem5d

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmuni 5567	. 2 ⊢ dom ∪ 𝐶 = ∪ 𝑔 ∈ 𝐶 dom 𝑔
2		ssel 3822	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑔 ∈ 𝐶 → 𝑔 ∈ 𝐵))
3		frrlem5.3	. . . . . 6 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
4	3	frrlem3 32322	. . . . 5 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)
5	2, 4	syl6 35	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝑔 ∈ 𝐶 → dom 𝑔 ⊆ 𝐴))
6	5	ralrimiv 3175	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ∀𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴)
7		iunss 4782	. . 3 ⊢ (∪ 𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴)
8	6, 7	sylibr 226	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∪ 𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴)
9	1, 8	syl5eqss 3875	1 ⊢ (𝐶 ⊆ 𝐵 → dom ∪ 𝐶 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658  ∃wex 1880   ∈ wcel 2166  {cab 2812  ∀wral 3118   ⊆ wss 3799  ∪ cuni 4659  ∪ ciun 4741   Fr wfr 5299   Se wse 5300  dom cdm 5343   ↾ cres 5345  Predcpred 5920   Fn wfn 6119  ‘cfv 6124  (class class class)co 6906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-iota 6087  df-fun 6126  df-fn 6127  df-fv 6132  df-ov 6909

This theorem is referenced by: (None)