Theorem iunss2 3927

Description: A subclass condition on the members of two indexed classes 𝐶(𝑥) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3836. (Contributed by NM, 9-Dec-2004.)

Assertion

Ref	Expression
iunss2	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem iunss2

Step	Hyp	Ref	Expression
1		ssiun 3924	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
2	1	ralimi 2538	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
3		iunss 3923	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
4	2, 3	sylibr 134	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4  ∀wral 2453  ∃wrex 2454   ⊆ wss 3127  ∪ ciun 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-in 3133  df-ss 3140  df-iun 3884

This theorem is referenced by:  iunxdif2  3930  rdgss  6374